Hyperbolic geometry and continued fraction theory...
Transcript of Hyperbolic geometry and continued fraction theory...
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry and continued fractiontheory I
Ian Short 9 February 2010
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Collaborators
Meira Hockman
Alan Beardon(University of Cambridge)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Project
The geometry of continued fractions
httpmathsorg ims25mathspresentationsphp
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fractions
K(an| bn) =a1
b1 +a2
b2 +a3
b3 +a4
b4 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Continued fraction convergence
a1
b1
a1
b1 +a2
b2
a1
b1 +a2
b2 +a3
b3
a1
b1 +a2
b2 +a3
b3 +a4
b4
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of e = 271828182845905 (Euler)
e = 2 +1
1 +1
2 +1
1 +1
1 +1
4 +1
1 +1
1 +1
6 +1
1 +1
1 +1
8 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Expansion of π = 314159265358979 (Lange)
π = 3 +12
6 +32
6 +52
6 +72
6 +92
6 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
A problematic example
1
2 +1
3 +1
1 +1
minus1 + middot middot middot
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius transformations
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
tn(infin) = 0
Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence again
Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Dealing with infin
Previously
1infin
= 0
Now
h(z) =1z
h(infin) = 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps
Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry
Simpler notation composition of maps rather thanalgebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three more reasons for using Mobius maps
There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than
algebraic manipulation
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Reminder
tn(z) =an
bn + z
Tn = t1 t2 middot middot middot tn
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Usual integer continued fractions
K(1| bn) bn isin N
tn(z) =1
bn + z
The modular group
Γ =z 7rarr az + b
cz + d a b c d isin Z adminus bc = 1
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Complex continued fractions
K(an| bn) an bn isin C
tn(z) =an
bn + z
The Mobius group
M =z 7rarr az + b
cz + d a b c d isin C adminus bc 6= 0
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
M generated by inversions
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Two aspects of M
Three-dimensional hyperbolic isometries Topological group
(and complete metric space)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Two aspects of M
Three-dimensional hyperbolic isometries
Topological group
(and complete metric space)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Two aspects of M
Three-dimensional hyperbolic isometries Topological group (and complete metric space)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Two aspects of M
Three-dimensional hyperbolic isometries
Topological group
(and complete metric space)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Two aspects of M
Three-dimensional hyperbolic isometries Topological group (and complete metric space)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Two aspects of M
Three-dimensional hyperbolic isometries Topological group (and complete metric space)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem I
Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
hyperbolic geometry
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
H3 = (x1 x2 x3) isin R3 x3 gt 0
Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Upper half-space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
(H3 ρ)
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Three-dimensional hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Graph representation of hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius action on hyperbolic space
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Isometry group
Isom(H3) =M
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Back to continued fractions
tn(infin) = 0 Tn(infin) = Tnminus1(0)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Suppose K(an| bn) converges
In other words suppose T1(0) T2(0) converges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Convergence
IfTn(0)rarr p
thenTn(j)rarr p
(Lorentzen Aebischer Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the hyperbolic metric
sinh 12ρ(x y) =
|xminus y|2radicx3y3
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j))
= ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))
= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)
sinh 12ρ(j tn(j)) =
|bn|2
ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin then
sumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Hyperbolic distance calculations
sinh 12ρ(Tnminus1(j) Tn(j)) =
|bn|2
Hence ifsum
n |bn| lt +infin thensumn
ρ(Tnminus1(j) Tn(j)) lt +infin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Cannot reach the boundary (Beardon)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Open Problem II
What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Toplogical groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Schematic diagram
continued fractions
Mobius maps
topological groups
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3
Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Stereographic projection
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
χ(w z) =2|w minus z|radic
1 + |w|2radic
1 + |z|2χ(winfin) =
2radic1 + |w|2
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The chordal metric
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
The supremum metric
χ0(f g) = supzisinCinfin
χ(f(z) g(z))
f g isinM
The metric of uniform convergence
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space
(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group
right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)
h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Mobius group
(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Recall the SternndashStolz Theorem
Theorem Ifsum
n |bn| converges then K(1| bn) diverges
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Key observation
If bn small then
tn(z) =1
bn + zsim h(z) =
1z
We must calculate χ0(tn h)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h)
= χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)
= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(tn h)
χ0(tn h) = χ0(htn I)= χ0(z + bn z)
= supzisinCinfin
2|bn|radic1 + |z|2
radic1 + |z + bn|2
le 2|bn|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2)
= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)
= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)
le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)
= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)
le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Calculate χ0(Tminus1n Tminus1
n+2)
χ0(Tminus1n Tminus1
n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin then
sumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
Ifsum
n |bn| lt +infin thensumn
χ0(Tminus1n Tminus1
n+2) lt +infin
Hence Tminus12nminus1 converges uniformly to a Mobius map f
Hence T2nminus1 converges uniformly to a Mobius map g
Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Conclusion
In summary ifsum
n |bn| lt +infin then there is a Mobius map gsuch that
T2nminus1 rarr g T2n rarr gh
Hence
T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-
Continued fractions Mobius maps Hyperbolic geometry Toplogical groups
Thank you
- Continued fractions
-
- Moumlbius maps
-
- Hyperbolic geometry
-
- Toplogical groups
-