Semigroups of M obius transformations Matthew Jacques Thursday 12 - PowerPoint PPT Presentation

Semigroups of M¨ obius transformations Matthew Jacques Thursday 12 th March 2015 - Joint work with Ian Short - Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 0 / 29

Introduction Contents Contents 1 M¨ obius transformations and hyperbolic geometry ■ M¨ obius transformations and their action inside the unit ball ■ The hyperbolic metric 2 Semigroups of M¨ obius transformations ■ Semigroups ■ Limit sets of M¨ obius semigroups ■ Examples 3 Composition sequences ■ Escaping and converging composition sequences ■ Examples 4 A Theorem on convergence Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 1 / 29

Introduction Contents Contents 1 M¨ obius transformations and hyperbolic geometry ■ M¨ obius transformations and their action inside the unit ball ■ The hyperbolic metric 2 Semigroups of M¨ obius transformations ■ Semigroups ■ Limit sets of M¨ obius semigroups ■ Examples 3 Composition sequences ■ Escaping and converging composition sequences ■ Examples 4 A Theorem on convergence Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 1 / 29

Introduction Contents Contents 1 M¨ obius transformations and hyperbolic geometry ■ M¨ obius transformations and their action inside the unit ball ■ The hyperbolic metric 2 Semigroups of M¨ obius transformations ■ Semigroups ■ Limit sets of M¨ obius semigroups ■ Examples 3 Composition sequences ■ Escaping and converging composition sequences ■ Examples 4 A Theorem on convergence Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 1 / 29

Introduction Contents Contents 1 M¨ obius transformations and hyperbolic geometry ■ M¨ obius transformations and their action inside the unit ball ■ The hyperbolic metric 2 Semigroups of M¨ obius transformations ■ Semigroups ■ Limit sets of M¨ obius semigroups ■ Examples 3 Composition sequences ■ Escaping and converging composition sequences ■ Examples 4 A Theorem on convergence Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 1 / 29

M¨ obius transformations and hyperbolic geometry M¨ obius transformations M¨ obius transformations M¨ obius transformations are the conformal automorphisms of ❜ ❈ = ❈ ❬ ❢✶❣ . That is the bijective functions on ❜ ❈ which preserve angles and their orientation. Each takes the form ✦ az + b z ✼� cz + d with a ❀ b ❀ c ❀ d ✷ ❈ and ad � bc ✻ = 0 Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 2 / 29

M¨ obius transformations and hyperbolic geometry M¨ obius transformations M¨ obius transformations M¨ obius transformations are the conformal automorphisms of ❜ ❈ = ❈ ❬ ❢✶❣ . That is the bijective functions on ❜ ❈ which preserve angles and their orientation. Each takes the form ✦ az + b z ✼� cz + d with a ❀ b ❀ c ❀ d ✷ ❈ and ad � bc ✻ = 0 Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 2 / 29

M¨ obius transformations and hyperbolic geometry M¨ obius transformations obius transformations acting on ❜ We consider the group ▼ of M¨ ❈ , which we identify with ❙ 2 . By decomposing the action of any given M¨ obius transformation into a composition of inversions in spheres orthogonal to ❙ 2 , the action of ▼ may be extended to a conformal action on ❘ 3 ❬ ❢✶❣ . In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 3 / 29

M¨ obius transformations and hyperbolic geometry M¨ obius transformations obius transformations acting on ❜ We consider the group ▼ of M¨ ❈ , which we identify with ❙ 2 . By decomposing the action of any given M¨ obius transformation into a composition of inversions in spheres orthogonal to ❙ 2 , the action of ▼ may be extended to a conformal action on ❘ 3 ❬ ❢✶❣ . In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 3 / 29

M¨ obius transformations and hyperbolic geometry M¨ obius transformations obius transformations acting on ❜ We consider the group ▼ of M¨ ❈ , which we identify with ❙ 2 . By decomposing the action of any given M¨ obius transformation into a composition of inversions in spheres orthogonal to ❙ 2 , the action of ▼ may be extended to a conformal action on ❘ 3 ❬ ❢✶❣ . In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 3 / 29

M¨ obius transformations and hyperbolic geometry M¨ obius transformations obius transformations acting on ❜ We consider the group ▼ of M¨ ❈ , which we identify with ❙ 2 . By decomposing the action of any given M¨ obius transformation into a composition of inversions in spheres orthogonal to ❙ 2 , the action of ▼ may be extended to a conformal action on ❘ 3 ❬ ❢✶❣ . In particular ▼ gives a conformal action on the closed unit ball, which it preserves. This extension is called the Poincar´ e extension . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 3 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric The hyperbolic metric, ✚ ( ✁ ❀ ✁ ) on ❇ 3 The hyperbolic metric ✚ on ❇ 3 is induced by the infinitesimal metric ❥ d x ❥ ds = 1 � ❥ x ❥ 2 ✿ • From any point inside ❇ 3 the distance to the ideal boundary, ❙ 2 , is infinite. • Geodesics are circular arcs which when extended land orthogonally on ❙ 2 . obius transformations that preserve ❇ 3 are exactly the set The group of M¨ of orientation preserving isometries of ( ❇ 3 ❀ ✚ ) . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 4 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric The hyperbolic metric, ✚ ( ✁ ❀ ✁ ) on ❇ 3 The hyperbolic metric ✚ on ❇ 3 is induced by the infinitesimal metric ❥ d x ❥ ds = 1 � ❥ x ❥ 2 ✿ • From any point inside ❇ 3 the distance to the ideal boundary, ❙ 2 , is infinite. • Geodesics are circular arcs which when extended land orthogonally on ❙ 2 . obius transformations that preserve ❇ 3 are exactly the set The group of M¨ of orientation preserving isometries of ( ❇ 3 ❀ ✚ ) . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 4 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric The hyperbolic metric, ✚ ( ✁ ❀ ✁ ) on ❇ 3 The hyperbolic metric ✚ on ❇ 3 is induced by the infinitesimal metric ❥ d x ❥ ds = 1 � ❥ x ❥ 2 ✿ • From any point inside ❇ 3 the distance to the ideal boundary, ❙ 2 , is infinite. • Geodesics are circular arcs which when extended land orthogonally on ❙ 2 . obius transformations that preserve ❇ 3 are exactly the set The group of M¨ of orientation preserving isometries of ( ❇ 3 ❀ ✚ ) . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 4 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric The hyperbolic metric, ✚ ( ✁ ❀ ✁ ) on ❇ 3 The hyperbolic metric ✚ on ❇ 3 is induced by the infinitesimal metric ❥ d x ❥ ds = 1 � ❥ x ❥ 2 ✿ • From any point inside ❇ 3 the distance to the ideal boundary, ❙ 2 , is infinite. • Geodesics are circular arcs which when extended land orthogonally on ❙ 2 . obius transformations that preserve ❇ 3 are exactly the set The group of M¨ of orientation preserving isometries of ( ❇ 3 ❀ ✚ ) . Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 4 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 5 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 5 / 29

M¨ obius transformations and hyperbolic geometry The hyperbolic metric Thursday 12 th March 2015 Matthew Jacques (The Open University) Semigroups of M¨ obius transformations 5 / 29