Knots and Dynamics
Étienne GhysUnité de Mathématiques Pures et Appliquées
CNRS - ENS Lyon
Lorenz equation (1963)
!
dx
dt=10 (y " x)
dy
dt= 28x " y " xz
dz
dt= xy "
8
3z
Birman-Williams: Periodic orbits are knots.
Topological description of the Lorenz attractor
41 Figure eight
Birman-Williams: Lorenz knots and links are very peculiar
• Lorenz knots are prime• Lorenz links are fibered• non trivial Lorenz links have positive signature
819 =T(4,3) 10124=T(5,3)
51Cinquefoil31 trefoil
10132
71
91
01 Unknot
Schwarzman-Sullivan- Thurston etc. One should think of a measure preserving vector field as « an asymptotic cycle »
!
k(T , x) = x" orbit
# $ # # "T (x)segment
# $ # # # x{ }
!
Flow " " limT#$"
1
Tk(T , x) dµ(x)%
Example : helicity as asymptotic linking number
+1
-1
Linking # = +1
Linking # = 0
Theorem (Arnold) Consider a flow in abounded domain in R3 preserving an ergodicprobability measure µ (not concentrated on aperiodic orbit). Then, for µ-almost every pairof points x1,x2, the following limit exists andis independent of x1,x2
!
Helicity = limT1 ,T2"#
1
T1T2Link(k(T1, x1),k(T2, x2))
Example : helicity as asymptotic linking number
Open question (Arnold): Is Helicity a topogical invariant ?
Let φ1t and φ2
t be two smooth flows preserving µ1 and µ1. Assume there is an orientation preserving homeomorphism h such that h o φ1
t = φ2t o h and h* µ1= µ2.
Does it follow that
Helicity(φ1t) = Helicity(φ2
t) ?
http://www.math.rug.nl/~veldman/movies/dns-large.mpg
Suspension: an area preserving diffeomorphimof the disk defines a volume preserving vectorfield in a solid torus.
Invariants on Diff (D2,area)?
x f(x)
Theorem (Banyaga): The Kernel of Cal is a simple group.
Theorem (Gambaudo-G): Cal( f ) = Helicity(Suspension f )
Theorem (Gambaudo-G): Cal is a topological invariant.
Corollary : Helicity is a topological invariant for thoseflows which are suspensions.
Theorem (Calabi): There is a non trivial homomorphism
!
Cal : Diff"(D
2,#D
2,area)$ R
A definition of Calabi’s invariant (Fathi)
!
f " Diff#(D
2,$D
2,area)
!
f0
= Id
!
f1
= f
!
f t (t " [0,1])
!
Cal( f ) = Vart= 0t=1"" Arg ft (x) # f t (y)( ) dx dy
Open question (Mather):
Is the group Homeo(D2, ∂D2, area) a simple group?
• Can one extend Cal to homeomorphisms?
• Good candidate for a normal subgroup: the group of « hameomorphisms » (Oh). Is it non trivial?
Abelian groupsor
SL(n,Z) for n>2)
No non trivial (Trauber,Burger-Monod)
More invariants on the group Diff(D2, ∂D2, area)?
No homomorphism besides Calabi’s…
Quasimorphisms
!
" :#$ R
!
" #1#2( ) $ " #
1( ) $ " #2( ) % Const
Homogeneous if
!
" # n( ) = n" #( )
Γ non abelian free group
Many non trivial homogeneousquasimorphisms (Gromov, …)
Theorem (Gambaudo-G) The vector space of homogeneousquasimorphisms on Diff(D2, ∂D2, area) is infinite dimensional.
One idea: use braids and quasimorphisms on braid groups.
!
(x1, x2,..., xn ) a b(x1, x2,..., xn ) " Bn#
$ % $ R
Average over n-tuples of points in the disk
Example: n=2. Calabi homomorphism.
More quasimorphims…
Theorem (Entov-Polterovich) :There exists a « Calabi quasimorphism »such thatwhen the support of f is in a disc D with area < 1/2 area(S2).
!
"( f ) =Cal( fD)
!
" :Diff0 (S2,area)# R
Theorem (Py) :If Σ is a compact orientable surface of genus g > 0,there is a « Calabi quasimorphism »such thatwhen the support of f is contained in some disc D ⊂ Σ.
!
"( f ) =Cal( fD)
!
" :Ham(#,area)$ R
Questions:
• Can one define similar invariants for volume preservingflows in 3-space, in the spirit of « Cal( f ) = Helicity(suspension f ) »?
• Does that produce topological invariants for smooth volume preserving flows?
• Higher dimensional topological invariants in symplecticdynamics?
• etc.
Example : the modular flow on SL(2,R)/SL(2,Z)
• Space of lattices in R2 of area 1
!
" # Z2$ R
2
!
area( R2/ ") = 1
Topology: SL(2,R)/SL(2,Z) is homeomorphic tothe complement of the trefoil knot in the 3-sphere
!
" # R2$Ca g2 ("),g3 (")( ) % C2 & g2
3 = 27g32{ }
!
g2 (") = 60 #$4
#%"$ 0{ }& % C
!
g3 (") = 140 #$6
#%"$ 0{ }& % C
!
area( R2/ ") = 1
g = 0g = 27 g32
3 23
S3
g = 02
Dynamics
!
" t (#) =et
0
0 e$t
%
& '
(
) * (#)On lattices
!
A " SL(2,Z)
• Conjugacy classes of hyperbolic elements in PSL(2,Z)• Closed geodesics on the modular surface D/PSL(2,Z)• Ideal classes in quadratic fields• Indefinite integral quadratic forms in two variables.• Continuous fractions etc.
!
A(Z2) = Z
2
!
" = P(Z2)
!
PAP"1
= ±et
0
0 e"t
#
$ %
&
' (
!
et
0
0 e"t
#
$ %
&
' ( ()) = )
Periodic orbits
Each hyperbolic matrix A in PSL(2,Z) defines aperiodic orbit in SL(2,R)/SL(2,Z), hence a closed curvekA in the complement of the trefoil knot.
Questions :
1) What kind of knots are the « modular knots » kA ?
2) Compute the linking number between kA and the trefoil.
Theorem: The linking number between kA andthe trefoil is equal to R(A) where R is the« Rademacher function ».
!
"a# + b
c# + d
$
% &
'
( )
24
= "(# )24 (c# + d)12
;a b
c d
$
% &
'
( ) * SL(2,Z)
Dedekind eta-function :
!
"(# ) = exp(i$# /12) 1% exp(2i$n# )( )n&1
' ; ((# ) > 0
This is a quasimorphism.
!
R : SL(2,Z)" Z
!
24(log")a# + b
c# + d
$
% &
'
( ) = 24(log")(# ) + 6 log(*(c# + d)2) + 2i+ R(A)
!
log(z) = log z + i Arg(z)
!
(g23" 27g3
2)(#) $ R
+ is a Seifert surface
« Proof » that R(A) = Linking(kA, )
is the trefoil knot
!
g2
3" 27g
3
2= 0
Jacobi proved that
!
(g23 " 27g3
2)(Z +#Z) = (2$ )12%(# )24
Theorem:Modular knots (and links) are the same as Lorenzknots (and links)
Step 1: find some template inside SL(2,R)/SL(2,Z) whichlooks like the Lorenz template.
Make it thicker by pushing along the unstable direction.
Look at « regular hexagonal lattices »
and lattices with horizontal rhombuses as fundamental domainwith angle between 60 and 120 degrees.
Step 2 : deform lattices to make them approachthe Lorenz template.
•From modular dynamics to Lorenz dynamics and vice versa?
• For instance, « modular explanation » of the fact that all Lorenz links are fibered?
Further developments?
Many thanks to Jos Leys !
Mathematical Imagery : http://www.josleys.com/
« A mathematical theory is not to be considered complete until youmade it so clear that you can explain it to the man you meet on thestreet »
« For what is clear and easily comprehended attracts and thecomplicated repels us »
D. Hilbert
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