Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos...
Transcript of Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos...
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Hyperbolic Geometry and Chaos in theComplex Plane
Mahan Mj,School of Mathematics,
Tata Institute of Fundamental Research.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 2: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/2.jpg)
The best mathematics uses the whole mind, embraces humansensibility, and is not at all limited to the small portion of ourbrains that calculates and manipulates symbols. Throughpursuing beauty we find truth, and where we find truth wediscover incredible beauty.– William Thurston (October 30, 1946 – August 21, 2012)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 3: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/3.jpg)
The best mathematics uses the whole mind, embraces humansensibility, and is not at all limited to the small portion of ourbrains that calculates and manipulates symbols. Throughpursuing beauty we find truth, and where we find truth wediscover incredible beauty.– William Thurston (October 30, 1946 – August 21, 2012)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 4: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/4.jpg)
The best mathematics uses the whole mind, embraces humansensibility, and is not at all limited to the small portion of ourbrains that calculates and manipulates symbols. Throughpursuing beauty we find truth, and where we find truth wediscover incredible beauty.– William Thurston (October 30, 1946 – August 21, 2012)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 5: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/5.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 6: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/6.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 7: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/7.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 8: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/8.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 9: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/9.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 10: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/10.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 11: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/11.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 12: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/12.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 13: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/13.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 14: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/14.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 15: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/15.jpg)
Hyperbolic metric spaces:
Triangles are thin: [a,b] ⊂ Nδ([a, c] ∪ [b, c]).For a tree, δ = 0.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 16: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/16.jpg)
Hyperbolic metric spaces:
Triangles are thin: [a,b] ⊂ Nδ([a, c] ∪ [b, c]).For a tree, δ = 0.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 17: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/17.jpg)
Hyperbolic metric spaces:
Triangles are thin: [a,b] ⊂ Nδ([a, c] ∪ [b, c]).For a tree, δ = 0.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 18: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/18.jpg)
Hyperbolic metric spaces:
Triangles are thin: [a,b] ⊂ Nδ([a, c] ∪ [b, c]).For a tree, δ = 0.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 19: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/19.jpg)
Note that limbs get smaller and smaller in order to fit in R2.
Accumulates to a Cantor set.ENTER FRACTALS.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 20: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/20.jpg)
Note that limbs get smaller and smaller in order to fit in R2.
Accumulates to a Cantor set.ENTER FRACTALS.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 21: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/21.jpg)
Note that limbs get smaller and smaller in order to fit in R2.
Accumulates to a Cantor set.ENTER FRACTALS.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 22: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/22.jpg)
Note that limbs get smaller and smaller in order to fit in R2.
Accumulates to a Cantor set.ENTER FRACTALS.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 23: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/23.jpg)
Groups = SymmetriesCayley graph of a (discrete) group G = 〈g1, · · · ,gk : r1, · · · , rs〉V = {g ∈ G}; E = {(a,b) : a−1b ∈ {g1, · · · ,gk}}.
F2
Boundaries of hyperbolic groups are fractals.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 24: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/24.jpg)
Groups = SymmetriesCayley graph of a (discrete) group G = 〈g1, · · · ,gk : r1, · · · , rs〉V = {g ∈ G}; E = {(a,b) : a−1b ∈ {g1, · · · ,gk}}.
F2
Boundaries of hyperbolic groups are fractals.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 25: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/25.jpg)
Groups = SymmetriesCayley graph of a (discrete) group G = 〈g1, · · · ,gk : r1, · · · , rs〉V = {g ∈ G}; E = {(a,b) : a−1b ∈ {g1, · · · ,gk}}.
F2
Boundaries of hyperbolic groups are fractals.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 26: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/26.jpg)
Groups = SymmetriesCayley graph of a (discrete) group G = 〈g1, · · · ,gk : r1, · · · , rs〉V = {g ∈ G}; E = {(a,b) : a−1b ∈ {g1, · · · ,gk}}.
F2
Boundaries of hyperbolic groups are fractals.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 27: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/27.jpg)
Groups = SymmetriesCayley graph of a (discrete) group G = 〈g1, · · · ,gk : r1, · · · , rs〉V = {g ∈ G}; E = {(a,b) : a−1b ∈ {g1, · · · ,gk}}.
F2
Boundaries of hyperbolic groups are fractals.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 28: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/28.jpg)
G = 〈a,b, c|a2,b2, c2, (ab)2, (bc)4, (ca)6〉
Reflections in a hyperbolic triangle with angles π2 ,
π4 ,
π6 .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 29: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/29.jpg)
G = 〈a,b, c|a2,b2, c2, (ab)2, (bc)4, (ca)6〉
Reflections in a hyperbolic triangle with angles π2 ,
π4 ,
π6 .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 30: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/30.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 31: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/31.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 32: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/32.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 33: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/33.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 34: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/34.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 35: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/35.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 36: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/36.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 37: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/37.jpg)
Relative problem:
H ⊂ G hyperbolic subgroup of a hyperbolic group.i : ΓH → ΓG inclusion of Cayley graphs.Does i extend to a continuous map between the fractalboundaries?Answer is "No" in this generality. (Baker-Riley 2013)But an analogous (and much more classical) problem ariseswhen a hyperbolic group acts by symmetries (isometries) on H3
– 3 dimensional hyperbolic space.H3 = {(x , y , z) : z > 0} equipped with metric ds2 = dx2+dy2+dz2
z2 .Boundary is C ∪ {∞}.If action is nice on H3, then geometer’s instinct tells us to take aquotient.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 38: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/38.jpg)
Z ⊕ Z acts on R2 to give torus S1 × S1.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 39: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/39.jpg)
Z ⊕ Z acts on R2 to give torus S1 × S1.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 40: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/40.jpg)
Mahan Mj Hyperbolic Geometry and Fractals
![Page 41: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/41.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 42: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/42.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 43: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/43.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 44: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/44.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 45: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/45.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 46: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/46.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
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Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
![Page 48: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/48.jpg)
Return to 3 dimensional problem.Discrete subgroup G of group of Mobius transformationsMob(C) = PSL2(C) = Isom(H3).Quotient: Fundamental group of a hyperbolic manifoldM3 = H3/G.S2 = C is the ‘ideal’ boundary of H3.Mob(C) is given by z → az+b
cz+d .
Mahan Mj Hyperbolic Geometry and Fractals
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Theorem:There is an exact dictionary between1) The dynamics of G on S2 = C, –Fractal.2) The geometry of M3 – Hyperbolic Geometry.
Mahan Mj Hyperbolic Geometry and Fractals
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Theorem:There is an exact dictionary between1) The dynamics of G on S2 = C, –Fractal.2) The geometry of M3 – Hyperbolic Geometry.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 51: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/51.jpg)
Theorem:There is an exact dictionary between1) The dynamics of G on S2 = C, –Fractal.2) The geometry of M3 – Hyperbolic Geometry.
Mahan Mj Hyperbolic Geometry and Fractals
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Theorem:There is an exact dictionary between1) The dynamics of G on S2 = C, –Fractal.2) The geometry of M3 – Hyperbolic Geometry.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 53: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/53.jpg)
Theorem:There is an exact dictionary between1) The dynamics of G on S2 = C, –Fractal.2) The geometry of M3 – Hyperbolic Geometry.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 54: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/54.jpg)
Theorem:There is an exact dictionary between1) The dynamics of G on S2 = C, –Fractal.2) The geometry of M3 – Hyperbolic Geometry.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 55: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/55.jpg)
Limit set ΛG = Set of accumulation points in C of G.o for some(any) o ∈ H3.Hence for the (2,4,6)−group or the double torus (octagonaltiling) group, limit set = round equatorial circle.The intrinsic boundary (a circle) embeds as a round circle in S2.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 56: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/56.jpg)
Limit set ΛG = Set of accumulation points in C of G.o for some(any) o ∈ H3.Hence for the (2,4,6)−group or the double torus (octagonaltiling) group, limit set = round equatorial circle.The intrinsic boundary (a circle) embeds as a round circle in S2.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 57: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/57.jpg)
Limit set ΛG = Set of accumulation points in C of G.o for some(any) o ∈ H3.Hence for the (2,4,6)−group or the double torus (octagonaltiling) group, limit set = round equatorial circle.The intrinsic boundary (a circle) embeds as a round circle in S2.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 58: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/58.jpg)
Limit set ΛG = Set of accumulation points in C of G.o for some(any) o ∈ H3.Hence for the (2,4,6)−group or the double torus (octagonaltiling) group, limit set = round equatorial circle.The intrinsic boundary (a circle) embeds as a round circle in S2.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 59: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/59.jpg)
Limit set ΛG = Set of accumulation points in C of G.o for some(any) o ∈ H3.Hence for the (2,4,6)−group or the double torus (octagonaltiling) group, limit set = round equatorial circle.The intrinsic boundary (a circle) embeds as a round circle in S2.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 60: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/60.jpg)
Limit set ΛG = Set of accumulation points in C of G.o for some(any) o ∈ H3.Hence for the (2,4,6)−group or the double torus (octagonaltiling) group, limit set = round equatorial circle.The intrinsic boundary (a circle) embeds as a round circle in S2.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 61: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/61.jpg)
Deform:
Mahan Mj Hyperbolic Geometry and Fractals
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Limit set is the locus of chaotic dynamics of the G−action onS2.i : ΓG → H3 sending g ∈ G to g.o ∈ H3.Does i extend to a continuous map between the circleboundary of G and its limit set?A continuous map as above (if it exists) is called aCannon-Thurston map.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 63: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/63.jpg)
Limit set is the locus of chaotic dynamics of the G−action onS2.i : ΓG → H3 sending g ∈ G to g.o ∈ H3.Does i extend to a continuous map between the circleboundary of G and its limit set?A continuous map as above (if it exists) is called aCannon-Thurston map.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 64: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/64.jpg)
Limit set is the locus of chaotic dynamics of the G−action onS2.i : ΓG → H3 sending g ∈ G to g.o ∈ H3.Does i extend to a continuous map between the circleboundary of G and its limit set?A continuous map as above (if it exists) is called aCannon-Thurston map.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 65: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/65.jpg)
Limit set is the locus of chaotic dynamics of the G−action onS2.i : ΓG → H3 sending g ∈ G to g.o ∈ H3.Does i extend to a continuous map between the circleboundary of G and its limit set?A continuous map as above (if it exists) is called aCannon-Thurston map.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 66: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/66.jpg)
Limit set is the locus of chaotic dynamics of the G−action onS2.i : ΓG → H3 sending g ∈ G to g.o ∈ H3.Does i extend to a continuous map between the circleboundary of G and its limit set?A continuous map as above (if it exists) is called aCannon-Thurston map.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 67: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/67.jpg)
Limit set is the locus of chaotic dynamics of the G−action onS2.i : ΓG → H3 sending g ∈ G to g.o ∈ H3.Does i extend to a continuous map between the circleboundary of G and its limit set?A continuous map as above (if it exists) is called aCannon-Thurston map.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 68: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/68.jpg)
Theorem(M-) There exist Cannon-Thurston maps for finitely generated(3d) Kleinian groups.
Theorem(M-) Connected limit sets of f.g. (3d) Kleinian groups are locallyconnected.
Second follows from first using a result of Anderson-Maskit.
Mahan Mj Hyperbolic Geometry and Fractals
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Theorem(M-) There exist Cannon-Thurston maps for finitely generated(3d) Kleinian groups.
Theorem(M-) Connected limit sets of f.g. (3d) Kleinian groups are locallyconnected.
Second follows from first using a result of Anderson-Maskit.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 70: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/70.jpg)
Theorem(M-) There exist Cannon-Thurston maps for finitely generated(3d) Kleinian groups.
Theorem(M-) Connected limit sets of f.g. (3d) Kleinian groups are locallyconnected.
Second follows from first using a result of Anderson-Maskit.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 71: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/71.jpg)
Theorem(M-) There exist Cannon-Thurston maps for finitely generated(3d) Kleinian groups.
Theorem(M-) Connected limit sets of f.g. (3d) Kleinian groups are locallyconnected.
Second follows from first using a result of Anderson-Maskit.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 72: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/72.jpg)
Theorem(M-) There exist Cannon-Thurston maps for finitely generated(3d) Kleinian groups.
Theorem(M-) Connected limit sets of f.g. (3d) Kleinian groups are locallyconnected.
Second follows from first using a result of Anderson-Maskit.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 73: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/73.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 74: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/74.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 75: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/75.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 76: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/76.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 77: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/77.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 78: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/78.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 79: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/79.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 80: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/80.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals
![Page 81: Hyperbolic Geometry and Chaos in the Complex Planemahan/ct_gen.pdf · Hyperbolic Geometry and Chaos in the Complex Plane Mahan Mj, School of Mathematics, Tata Institute of Fundamental](https://reader033.fdocuments.in/reader033/viewer/2022053002/5f06566c7e708231d4177d62/html5/thumbnails/81.jpg)
TheoremEnding Lamination Theorem: (Brock-Canary-Minsky)Asymptotic topology (at infinity) of M determines geometry ofM.
Theorem(M-) The asymptotic topology is determined by theCannon-Thurston map.
TheoremChaotic dynamics on boundary determines and is determinedby the geometry of M.
Mahan Mj Hyperbolic Geometry and Fractals