mathematical billiards, flat surfaces and...
TRANSCRIPT
![Page 1: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/1.jpg)
Prospects in Mathematics, Edinburgh, 17-18 December 2010
Mathematical Billiards,Flat Surfacesand Dynamics
Corinna Ulcigrai
![Page 2: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/2.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 3: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/3.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 4: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/4.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 5: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/5.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 6: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/6.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 7: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/7.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 8: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/8.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 9: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/9.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 10: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/10.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 11: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/11.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 12: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/12.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 13: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/13.jpg)
Mathematical Billiards
A mathematical billiard consists of:
a billiard table: P ⊂ R2;
a ball: point particle with no-mass;
Billiard motion: straight lines, unit speed,elastic reflections at boundary:angle of incidende = angle of reflection.
No friction. Consider trajectories whichnever hit a pocket: trajectories and motionare infinite.
Tables can have various shapes:
![Page 14: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/14.jpg)
An L-shape billiard in real life:
[Credit: Photo courtesy of Moon Duchin]
![Page 15: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/15.jpg)
Why to study mathematical billiards?
Mathematical billiards arise naturally in many problems in physics, e.g.:
I Two masses on a rod:
(billiard in a triangle)
I Lorentz Gas
(Sinaibilliard)
![Page 16: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/16.jpg)
Why to study mathematical billiards?
Mathematical billiards arise naturally in many problems in physics, e.g.:
I Two masses on a rod:
(billiard in a triangle)
I Lorentz Gas
(Sinaibilliard)
![Page 17: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/17.jpg)
Why to study mathematical billiards?
Mathematical billiards arise naturally in many problems in physics, e.g.:
I Two masses on a rod:
(billiard in a triangle)
I Lorentz Gas
(Sinaibilliard)
![Page 18: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/18.jpg)
Why to study mathematical billiards?
Mathematical billiards arise naturally in many problems in physics, e.g.:
I Two masses on a rod:
(billiard in a triangle)
I Lorentz Gas
(Sinaibilliard)
![Page 19: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/19.jpg)
Why to study mathematical billiards?
Mathematical billiards arise naturally in many problems in physics, e.g.:
I Two masses on a rod:
(billiard in a triangle)
I Lorentz Gas
(Sinaibilliard)
![Page 20: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/20.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 21: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/21.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 22: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/22.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 23: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/23.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 24: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/24.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 25: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/25.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 26: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/26.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 27: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/27.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 28: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/28.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 29: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/29.jpg)
Billiards as Dynamical Systems
Mathematical billiards are an example of a dynamical system,that is a system that evolves in time.
Usually dynamical systems are chaotic and one is interested indetermining the asymptotic behaviour, or long-time evolution of thetrajectories.
Basic questions are:
I Are the periodictrajectories?
I Are trajectories dense?
I If a trajectory is dense,is it equidistributed?
![Page 30: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/30.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 31: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/31.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 32: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/32.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 33: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/33.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 34: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/34.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 35: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/35.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 36: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/36.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 37: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/37.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 38: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/38.jpg)
Shape of billiards and areas of dynamics
Integrable billiards(convex boundary)
many periodic orbits
Hamiltonian Dynamicsvariational methods
Polygonal billiards(flat boundary)
Teichmuller Dynamicsvery active area(Kontsevitch,McMullen, Yoccoz)
Hyperbolic billiards(convex boundary)
Hyperbolic dynamicsvery chaotic
![Page 39: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/39.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 40: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/40.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 41: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/41.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 42: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/42.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 43: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/43.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 44: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/44.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 45: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/45.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 46: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/46.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 47: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/47.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 48: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/48.jpg)
Unfolding a billiard trajectory: the square
From a billiard to a surface (Katok-Zemlyakov construction):
Instead than reflecting thetrajectory, REFLECT theTABLE!
Four copies are enough.
Glueing opposite sides onegets a torus
![Page 49: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/49.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 50: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/50.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 51: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/51.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 52: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/52.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 53: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/53.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 54: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/54.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 55: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/55.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 56: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/56.jpg)
Unfolding a billiard trajectory: higher genus
The unfolding construction works for any rational billiard, that is anypolygonal billiard with angles of the form π pi
qi.
E.g.: billiard in a triangle with π8 ,
3π8 ,
π2 angles.
Unfolding, one gets linear flow in the regular octagon.If we glue opposite sides, this is a surface of genus 2. Linear trajectorieshave one saddle singularity.
![Page 57: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/57.jpg)
Flows on surfaces
We saw that billiard in rational polygons give rise to flows on surfaces.
More in general, trajectories of flows on surfaces also arise from solutionsof differential equations:
The same trajectories can beobtained as local solutions ofHamiltonian equations:{ ∂x
∂t = ∂H∂y
∂y∂t = −∂H
∂x
ft(p) trajectory of p as t grows.
[Another motivation from solid state physics: locally Hamiltonian flows on
surfaces describe the motion of an electron under a magnetic filed on the Fermi
energy level surface in the semi-classical limit (Novikov).]
These flows ft : X → X preserves the area. What are the dynamicalproperties of the trajectories?
Next: two of my recent works: mixing in area-preserving flows onsurfaces and symbolic coding of trajectories in the octagon.
![Page 58: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/58.jpg)
Flows on surfaces
We saw that billiard in rational polygons give rise to flows on surfaces.
More in general, trajectories of flows on surfaces also arise from solutionsof differential equations:
The same trajectories can beobtained as local solutions ofHamiltonian equations:{ ∂x
∂t = ∂H∂y
∂y∂t = −∂H
∂x
ft(p) trajectory of p as t grows.
[Another motivation from solid state physics: locally Hamiltonian flows on
surfaces describe the motion of an electron under a magnetic filed on the Fermi
energy level surface in the semi-classical limit (Novikov).]
These flows ft : X → X preserves the area. What are the dynamicalproperties of the trajectories?
Next: two of my recent works: mixing in area-preserving flows onsurfaces and symbolic coding of trajectories in the octagon.
![Page 59: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/59.jpg)
Flows on surfaces
We saw that billiard in rational polygons give rise to flows on surfaces.
More in general, trajectories of flows on surfaces also arise from solutionsof differential equations:
The same trajectories can beobtained as local solutions ofHamiltonian equations:{ ∂x
∂t = ∂H∂y
∂y∂t = −∂H
∂x
ft(p) trajectory of p as t grows.
[Another motivation from solid state physics: locally Hamiltonian flows on
surfaces describe the motion of an electron under a magnetic filed on the Fermi
energy level surface in the semi-classical limit (Novikov).]
These flows ft : X → X preserves the area. What are the dynamicalproperties of the trajectories?
Next: two of my recent works: mixing in area-preserving flows onsurfaces and symbolic coding of trajectories in the octagon.
![Page 60: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/60.jpg)
Flows on surfaces
We saw that billiard in rational polygons give rise to flows on surfaces.
More in general, trajectories of flows on surfaces also arise from solutionsof differential equations:
The same trajectories can beobtained as local solutions ofHamiltonian equations:{ ∂x
∂t = ∂H∂y
∂y∂t = −∂H
∂x
ft(p) trajectory of p as t grows.
[Another motivation from solid state physics: locally Hamiltonian flows on
surfaces describe the motion of an electron under a magnetic filed on the Fermi
energy level surface in the semi-classical limit (Novikov).]
These flows ft : X → X preserves the area. What are the dynamicalproperties of the trajectories?
Next: two of my recent works: mixing in area-preserving flows onsurfaces and symbolic coding of trajectories in the octagon.
![Page 61: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/61.jpg)
Flows on surfaces
We saw that billiard in rational polygons give rise to flows on surfaces.
More in general, trajectories of flows on surfaces also arise from solutionsof differential equations:
The same trajectories can beobtained as local solutions ofHamiltonian equations:{ ∂x
∂t = ∂H∂y
∂y∂t = −∂H
∂x
ft(p) trajectory of p as t grows.
[Another motivation from solid state physics: locally Hamiltonian flows on
surfaces describe the motion of an electron under a magnetic filed on the Fermi
energy level surface in the semi-classical limit (Novikov).]
These flows ft : X → X preserves the area. What are the dynamicalproperties of the trajectories?
Next: two of my recent works: mixing in area-preserving flows onsurfaces and symbolic coding of trajectories in the octagon.
![Page 62: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/62.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 63: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/63.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 64: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/64.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 65: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/65.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 66: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/66.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 67: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/67.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 68: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/68.jpg)
Definition of Mixingft(p) describe the trajectory of p as t grows, each ft : X → X flowpreserves the area. Assume that the total area is 1.
Let A be a subset of the space X . Flow points in A for time t: doesft(A) spreads uniformly? E.g.
DefinitionThe flow ft is mixing if for any two subsets A,B
Area(ft(A) ∩ B)
Area(A)t→∞−−−→ Area(B).
Question (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
![Page 69: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/69.jpg)
Mixing in flows on surfacesQuestion (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
Theorem (U’07)A typical locally Hamiltonianflow on a surface that hastraps is mixing in thecomplement of the traps.
Theorem (U’09)A typical locally Hamiltonianflow on a surface that has onlysaddles (in particular, no traps)is not mixing.
Tools: dynamical systems, analysis (growth of certain functions),geometry (dynamics on the space of all surfaces), number theory andcombinatorics (Diophantine conditions)
![Page 70: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/70.jpg)
Mixing in flows on surfacesQuestion (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
Theorem (U’07)A typical locally Hamiltonianflow on a surface that hastraps is mixing in thecomplement of the traps.
Theorem (U’09)A typical locally Hamiltonianflow on a surface that has onlysaddles (in particular, no traps)is not mixing.
Tools: dynamical systems, analysis (growth of certain functions),geometry (dynamics on the space of all surfaces), number theory andcombinatorics (Diophantine conditions)
![Page 71: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/71.jpg)
Mixing in flows on surfacesQuestion (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
Theorem (U’07)A typical locally Hamiltonianflow on a surface that hastraps is mixing in thecomplement of the traps.
Theorem (U’09)A typical locally Hamiltonianflow on a surface that has onlysaddles (in particular, no traps)is not mixing.
Tools: dynamical systems, analysis (growth of certain functions),geometry (dynamics on the space of all surfaces), number theory andcombinatorics (Diophantine conditions)
![Page 72: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/72.jpg)
Mixing in flows on surfacesQuestion (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
Theorem (U’07)A typical locally Hamiltonianflow on a surface that hastraps is mixing in thecomplement of the traps.
Theorem (U’09)A typical locally Hamiltonianflow on a surface that has onlysaddles (in particular, no traps)is not mixing.
Tools: dynamical systems, analysis (growth of certain functions),geometry (dynamics on the space of all surfaces), number theory andcombinatorics (Diophantine conditions)
![Page 73: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/73.jpg)
Mixing in flows on surfacesQuestion (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
Theorem (U’07)A typical locally Hamiltonianflow on a surface that hastraps is mixing in thecomplement of the traps.
Theorem (U’09)A typical locally Hamiltonianflow on a surface that has onlysaddles (in particular, no traps)is not mixing.
Tools: dynamical systems, analysis (growth of certain functions),geometry (dynamics on the space of all surfaces), number theory andcombinatorics (Diophantine conditions)
![Page 74: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/74.jpg)
Mixing in flows on surfacesQuestion (Arnold, 80s) are locally Hamiltonian flows on surfaces mixing?
Theorem (U’07)A typical locally Hamiltonianflow on a surface that hastraps is mixing in thecomplement of the traps.
Theorem (U’09)A typical locally Hamiltonianflow on a surface that has onlysaddles (in particular, no traps)is not mixing.
Tools: dynamical systems, analysis (growth of certain functions),geometry (dynamics on the space of all surfaces), number theory andcombinatorics (Diophantine conditions)
![Page 75: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/75.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 76: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/76.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 77: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/77.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 78: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/78.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 79: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/79.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 80: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/80.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
55
6
6
77
8
8
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 81: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/81.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
55
6
6
77
8
8
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 82: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/82.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
55
6
6
77
8
8
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 83: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/83.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 84: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/84.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 85: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/85.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 86: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/86.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 87: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/87.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
55
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 88: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/88.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
55
6
6
7
A
C C
D B
B D
A
1
1
2
2
3
3
4
4
55
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 89: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/89.jpg)
Symbolic coding of trajectories in the octagon
Consider a regular octagon (or more ingeneral regular polygon with 2n sides).Glue opposite sides.Label pairs of sides by {A,B,C ,D}.
Let f θt be the linear flow in direction θ:trajectories which do not hit singularitiesare straight lines in direction θ.
D B
B D
A
1
1
2
2
3
3
4
4
55
6
6
7
A
C C
D B
B D
A
1
1
2
2
3
3
4
4
55
A
C C
D B
B D
A
1
1
2
2
3
3
4
4
55
6
6
77
A
C C
Definition (Cutting sequence)The cutting sequence in {A,B,C ,D}Z that codes a bi-infinite lineartrajectory is the sequence of labels of sides hit by the trajectory.
E.g.: The cutting sequence of the trajectory in the example contains:. . . A B B A C D . . .
Problem:Which sequences in {A,B,C ,D}Z are cutting sequences?
![Page 90: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/90.jpg)
A classical case: Sturmian sequences
Consider the special case in which the polygon is a square.
A
A
B B
In this case the cutting sequences are Sturmian sequences:
I Sturmian sequences correspond to the sequence of horizontal (letterA) and vertical (letter B) sides crossed by a line in direction θ in asquare grid: . . . A B A B B A B. . .
I Sturmian sequences appear in many areas of mathematics, e.g.in Computer Science - smallest possible complexity;in Number Theory - related to tan θ = 1
a1+1
a2+ 1a3+...
(continued fractions)
![Page 91: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/91.jpg)
A classical case: Sturmian sequencesConsider the special case in which the polygon is a square.
A
A
B B
A
AB
BB
A
B
In this case the cutting sequences are Sturmian sequences:
I Sturmian sequences correspond to the sequence of horizontal (letterA) and vertical (letter B) sides crossed by a line in direction θ in asquare grid: . . . A B A B B A B. . .
I Sturmian sequences appear in many areas of mathematics, e.g.in Computer Science - smallest possible complexity;in Number Theory - related to tan θ = 1
a1+1
a2+ 1a3+...
(continued fractions)
![Page 92: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/92.jpg)
A classical case: Sturmian sequencesConsider the special case in which the polygon is a square.
A
A
B B
A
AB
BB
A
B
In this case the cutting sequences are Sturmian sequences:
I Sturmian sequences correspond to the sequence of horizontal (letterA) and vertical (letter B) sides crossed by a line in direction θ in asquare grid: . . . A B A B B A B. . .
I Sturmian sequences appear in many areas of mathematics, e.g.in Computer Science - smallest possible complexity;in Number Theory - related to tan θ = 1
a1+1
a2+ 1a3+...
(continued fractions)
![Page 93: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/93.jpg)
A classical case: Sturmian sequencesConsider the special case in which the polygon is a square.
A
A
B B
A
AB
BB
A
B
In this case the cutting sequences are Sturmian sequences:
I Sturmian sequences correspond to the sequence of horizontal (letterA) and vertical (letter B) sides crossed by a line in direction θ in asquare grid: . . . A B A B B A B. . .
I Sturmian sequences appear in many areas of mathematics, e.g.in Computer Science - smallest possible complexity;in Number Theory - related to tan θ = 1
a1+1
a2+ 1a3+...
(continued fractions)
![Page 94: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/94.jpg)
A classical case: Sturmian sequencesConsider the special case in which the polygon is a square.
A
A
B B
A
AB
BB
A
B
In this case the cutting sequences are Sturmian sequences:
I Sturmian sequences correspond to the sequence of horizontal (letterA) and vertical (letter B) sides crossed by a line in direction θ in asquare grid: . . . A B A B B A B. . .
I Sturmian sequences appear in many areas of mathematics, e.g.in Computer Science - smallest possible complexity;in Number Theory - related to tan θ = 1
a1+1
a2+ 1a3+...
(continued fractions)
![Page 95: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/95.jpg)
A classical case: Sturmian sequencesConsider the special case in which the polygon is a square.
A
A
B B
A
AB
BB
A
B
In this case the cutting sequences are Sturmian sequences:
I Sturmian sequences correspond to the sequence of horizontal (letterA) and vertical (letter B) sides crossed by a line in direction θ in asquare grid: . . . A B A B B A B. . .
I Sturmian sequences appear in many areas of mathematics, e.g.in Computer Science - smallest possible complexity;in Number Theory - related to tan θ = 1
a1+1
a2+ 1a3+...
(continued fractions)
![Page 96: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/96.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
D B
B D
A
A
C C
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 97: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/97.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
D B
B D
A
A
C C
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 98: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/98.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
CA D B
A D
D B
B D
A
A
C C
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 99: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/99.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
CA D B
A D
O1 O2O3O4
O5O6
O8O7
A DADDefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 100: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/100.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
CA D B
A D B
O1O3O4
O5O6
O8O7
A O2D DA BDB
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 101: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/101.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
CA D B
CA D B
O1 O2O3O4
O5O6
O8O7 B
CBD
BCD
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 102: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/102.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
CA D B
CA D B
O1 O2O3O4
O5O6
O8O7C
CCCB B
C
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 103: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/103.jpg)
Admissible sequencesFirst restriction: only certain pairs ofconsecutive letters (transitions) can occurr.
E.g. if θ ∈[0, π8
], the transitions which can
appear correspond to the arrows in thediagram:
0
CA D B
CA D B
D B
B D
A
A
C C
DefinitionA sequence in {A,B,C ,D}Z is admissible if it uses only the arrows on
this diagram or in one corresponding to another sector[
kπ8 ,
(k+1)π8
]of
directions.
.
![Page 104: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/104.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 105: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/105.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,
. . . A
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 106: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/106.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 107: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/107.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A B
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 108: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/108.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A B A
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 109: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/109.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A B A C D D C . . .
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 110: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/110.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A B A C D D C . . .
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 111: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/111.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A B A C D D C . . .
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 112: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/112.jpg)
Derivation and characterization of cutting sequences
DefinitionA letter is sandwitched if it is preceeded and followed by the same letter,e.g. in D B B C B A A D the letter C is sandwitched between to Bs.
The derived sequence of a cutting sequence is obtained by keeping onlythe letters that are sandwitched and erasing the other letters, e.g.
. . . D A D B C C B C C B D A D B C B D B D B C B D . . . ,. . . A B A C D D C . . .
DefinitionA sequence in {A,B,C ,D}Z is derivable if it is admissible and its derivedsequence is again admissible.
Theorem (Smillie- U ’08)An octagon cutting sequence is infinitly derivable.With an additional condition, it becomes an iff (full combinatorialcharacterization of cutting sequences), analogous to the characterizationof Sturmian sequences using continued fractions.
![Page 113: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/113.jpg)
Renormalization and Teichmuller dynamics
The ideas in the proofs of the characterization of cutting sequences areinspired by tools in Teichmuller dynamics.
SL(2,R) affine deformationsof the octagon
VO < SL(2,R) Veech groupof the octagon
Teichmueller diskSL(2,R)V (O)
affine deformationsaffini automorphisms
Given θ, follow gθt geodesicray in direction θto choose moves on a tree ofaffine deformations
![Page 114: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/114.jpg)
Ergodic Theory and Dynamical Sytems in Bristol
The ergodic theory and dynamical systmes group in Bristol:
JensMarklov
AlexGorodnik
CorinnaUlcigrai
ThomasJordan
CarlDettmann
Post-docs: Felipe Ramirez, Shirali Kadyrov,Edward Crane, Alan Haynes (Heilbronn fellows)
Postgraduate students: Emek Demirci, Orestis Georgiou, MaximKirsebom, Riz Rahman, Henry Reeve, Oliver Sargent
Related groups: Number theory, Mathematical Physics, Quantum Chaos
Furthermore: Ergodic Theory and Dynamical Systems Seminar, TaughtCourse Center (with Bath, Imperial, Oxford, Warwick), Heibronn Institute
![Page 115: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/115.jpg)
Ergodic Theory and Dynamical Sytems in Bristol
The ergodic theory and dynamical systmes group in Bristol:
JensMarklov
AlexGorodnik
CorinnaUlcigrai
ThomasJordan
CarlDettmann
Post-docs: Felipe Ramirez, Shirali Kadyrov,Edward Crane, Alan Haynes (Heilbronn fellows)
Postgraduate students: Emek Demirci, Orestis Georgiou, MaximKirsebom, Riz Rahman, Henry Reeve, Oliver Sargent
Related groups: Number theory, Mathematical Physics, Quantum Chaos
Furthermore: Ergodic Theory and Dynamical Systems Seminar, TaughtCourse Center (with Bath, Imperial, Oxford, Warwick), Heibronn Institute
![Page 116: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/116.jpg)
Ergodic Theory and Dynamical Sytems in Bristol
The ergodic theory and dynamical systmes group in Bristol:
JensMarklov
AlexGorodnik
CorinnaUlcigrai
ThomasJordan
CarlDettmann
Post-docs: Felipe Ramirez, Shirali Kadyrov,Edward Crane, Alan Haynes (Heilbronn fellows)
Postgraduate students: Emek Demirci, Orestis Georgiou, MaximKirsebom, Riz Rahman, Henry Reeve, Oliver Sargent
Related groups: Number theory, Mathematical Physics, Quantum Chaos
Furthermore: Ergodic Theory and Dynamical Systems Seminar, TaughtCourse Center (with Bath, Imperial, Oxford, Warwick), Heibronn Institute
![Page 117: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/117.jpg)
Ergodic Theory and Dynamical Sytems in Bristol
The ergodic theory and dynamical systmes group in Bristol:
JensMarklov
AlexGorodnik
CorinnaUlcigrai
ThomasJordan
CarlDettmann
Post-docs: Felipe Ramirez, Shirali Kadyrov,Edward Crane, Alan Haynes (Heilbronn fellows)
Postgraduate students: Emek Demirci, Orestis Georgiou, MaximKirsebom, Riz Rahman, Henry Reeve, Oliver Sargent
Related groups: Number theory, Mathematical Physics, Quantum Chaos
Furthermore: Ergodic Theory and Dynamical Systems Seminar, TaughtCourse Center (with Bath, Imperial, Oxford, Warwick), Heibronn Institute
![Page 118: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/118.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Jens Marklov
Dynamics, Number theory and quantum chaos
E.g. Lorentz Gas
Billiard with periodic scatterers ofradius ε.
I What is the limit of the lenghtof free flights as ε→ 0?
I Which equations (stochasticprocess) describe trajectories asε→ 0?
Tools: dynamical systems (flows onhomogeneous spaces) and numbertheory
![Page 119: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/119.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Jens Marklov
Dynamics, Number theory and quantum chaos
E.g. Lorentz Gas
Billiard with periodic scatterers ofradius ε.
I What is the limit of the lenghtof free flights as ε→ 0?
I Which equations (stochasticprocess) describe trajectories asε→ 0?
Tools: dynamical systems (flows onhomogeneous spaces) and numbertheory
![Page 120: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/120.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Jens Marklov
Dynamics, Number theory and quantum chaos
E.g. Lorentz Gas
Billiard with periodic scatterers ofradius ε.
I What is the limit of the lenghtof free flights as ε→ 0?
I Which equations (stochasticprocess) describe trajectories asε→ 0?
Tools: dynamical systems (flows onhomogeneous spaces) and numbertheory
![Page 121: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/121.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Jens Marklov
Dynamics, Number theory and quantum chaos
E.g. Lorentz Gas
Billiard with periodic scatterers ofradius ε.
I What is the limit of the lenghtof free flights as ε→ 0?
I Which equations (stochasticprocess) describe trajectories asε→ 0?
Tools: dynamical systems (flows onhomogeneous spaces) and numbertheory
![Page 122: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/122.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Jens Marklov
Dynamics, Number theory and quantum chaos
E.g. Lorentz Gas
Billiard with periodic scatterers ofradius ε.
I What is the limit of the lenghtof free flights as ε→ 0?
I Which equations (stochasticprocess) describe trajectories asε→ 0?
Tools: dynamical systems (flows onhomogeneous spaces) and numbertheory
![Page 123: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/123.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Jens Marklov
Dynamics, Number theory and quantum chaos
E.g. Lorentz Gas
Billiard with periodic scatterers ofradius ε.
I What is the limit of the lenghtof free flights as ε→ 0?
I Which equations (stochasticprocess) describe trajectories asε→ 0?
Tools: dynamical systems (flows onhomogeneous spaces) and numbertheory
![Page 124: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/124.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 125: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/125.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 126: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/126.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 127: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/127.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 128: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/128.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 129: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/129.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 130: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/130.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 131: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/131.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Alex Gorodnik
Dynamics and Number theory
E.g. Counting Rational points on quadratic surfaces
Consider a quadratic equationQ(x1, x2, x3) = k .
I Consider rational solutions, that is
Q(
p1
q ,p2
q ,p3
q
)= k .
I How many rational solutions withdenominator q ≤ Q, up to a givenheight H? how do they grow as Q andH grow?
I (similar questions on homogeneousvarieties?)
Tools: dynamical systems (ergodic theorems) to study asymptotics
![Page 132: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/132.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Thomas Jordan
Dimension theory of dynamical systems(fractal sets and Hausdorff dimension)
E.g. Consider the fractal Bedford-McMullen carpet.
The (Hausdorff) fractal
dimension is log(1+2(2log 3/ log 5))log 3
(approximately 1.308 )
I Q: What is the (Hausdorff)dimension of theorthogonal projections indifferent directions?
I A: 1 except for vertical projection when the dimension is log 4/ log 5.[Joint work with Ferguson (Warwick) and Schmerkin (Manchester)]
![Page 133: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/133.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Thomas Jordan
Dimension theory of dynamical systems(fractal sets and Hausdorff dimension)
E.g. Consider the fractal Bedford-McMullen carpet.
The (Hausdorff) fractal
dimension is log(1+2(2log 3/ log 5))log 3
(approximately 1.308 )
I Q: What is the (Hausdorff)dimension of theorthogonal projections indifferent directions?
I A: 1 except for vertical projection when the dimension is log 4/ log 5.[Joint work with Ferguson (Warwick) and Schmerkin (Manchester)]
![Page 134: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/134.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Thomas Jordan
Dimension theory of dynamical systems(fractal sets and Hausdorff dimension)
E.g. Consider the fractal Bedford-McMullen carpet.
The (Hausdorff) fractal
dimension is log(1+2(2log 3/ log 5))log 3
(approximately 1.308 )
I Q: What is the (Hausdorff)dimension of theorthogonal projections indifferent directions?
I A: 1 except for vertical projection when the dimension is log 4/ log 5.[Joint work with Ferguson (Warwick) and Schmerkin (Manchester)]
![Page 135: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/135.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Thomas Jordan
Dimension theory of dynamical systems(fractal sets and Hausdorff dimension)
E.g. Consider the fractal Bedford-McMullen carpet.
The (Hausdorff) fractal
dimension is log(1+2(2log 3/ log 5))log 3
(approximately 1.308 )
I Q: What is the (Hausdorff)dimension of theorthogonal projections indifferent directions?
I A: 1 except for vertical projection when the dimension is log 4/ log 5.[Joint work with Ferguson (Warwick) and Schmerkin (Manchester)]
![Page 136: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/136.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Carl Dettmann
Dynamical systems, billiards and simulations
E.g. Escape from billiards with holes
Consider a billiard with a holes.
I What is the escape rate of trajectories?
I What is the asymptotics as the holesize → 0?
I In the case of the circle with two holes:the Riemann-Hypotheses is equivalent tothe exact asymptotics in the small holelimit.
![Page 137: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/137.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Carl Dettmann
Dynamical systems, billiards and simulations
E.g. Escape from billiards with holes
Consider a billiard with a holes.
I What is the escape rate of trajectories?
I What is the asymptotics as the holesize → 0?
I In the case of the circle with two holes:the Riemann-Hypotheses is equivalent tothe exact asymptotics in the small holelimit.
![Page 138: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/138.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Carl Dettmann
Dynamical systems, billiards and simulations
E.g. Escape from billiards with holes
Consider a billiard with a holes.
I What is the escape rate of trajectories?
I What is the asymptotics as the holesize → 0?
I In the case of the circle with two holes:the Riemann-Hypotheses is equivalent tothe exact asymptotics in the small holelimit.
![Page 139: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/139.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Carl Dettmann
Dynamical systems, billiards and simulations
E.g. Escape from billiards with holes
Consider a billiard with a holes.
I What is the escape rate of trajectories?
I What is the asymptotics as the holesize → 0?
I In the case of the circle with two holes:the Riemann-Hypotheses is equivalent tothe exact asymptotics in the small holelimit.
![Page 140: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/140.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Carl Dettmann
Dynamical systems, billiards and simulations
E.g. Escape from billiards with holes
Consider a billiard with a holes.
I What is the escape rate of trajectories?
I What is the asymptotics as the holesize → 0?
I In the case of the circle with two holes:the Riemann-Hypotheses is equivalent tothe exact asymptotics in the small holelimit.
![Page 141: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/141.jpg)
Ergodic Theory and Dynamical Systems in Bristol:
Carl Dettmann
Dynamical systems, billiards and simulations
E.g. Escape from billiards with holes
Consider a billiard with a holes.
I What is the escape rate of trajectories?
I What is the asymptotics as the holesize → 0?
I In the case of the circle with two holes:the Riemann-Hypotheses is equivalent tothe exact asymptotics in the small holelimit.
![Page 142: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/142.jpg)
Ergodic Theory Network
Bristol is part of the UK network of One Day Ergodic Theory Meetings:
which includes:
I Liverpool University
I Manchester University
I Queen Mary
I Surrey University
I Warwick University
All these UK departments (but not only!) have active staff doingresearch in dynamical systems and ergodic theory.
![Page 143: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/143.jpg)
Ergodic Theory Network
Bristol is part of the UK network of One Day Ergodic Theory Meetings:
which includes:I Liverpool UniversityI Manchester UniversityI Queen MaryI Surrey UniversityI Warwick University
All these UK departments (but not only!) have active staff doingresearch in dynamical systems and ergodic theory.
![Page 144: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/144.jpg)
Ergodic Theory Network
Bristol is part of the UK network of One Day Ergodic Theory Meetings:
which includes:
I Liverpool University
I Manchester University
I Queen Mary
I Surrey University
I Warwick University
All these UK departments (but not only!) have active staff doingresearch in dynamical systems and ergodic theory.
![Page 145: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/145.jpg)
Ergodic Theory Network
Bristol is part of the UK network of One Day Ergodic Theory Meetings:
which includes:
I Liverpool University
I Manchester University
I Queen Mary
I Surrey University
I Warwick University
All these UK departments (but not only!) have active staff doingresearch in dynamical systems and ergodic theory.
![Page 146: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/146.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 147: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/147.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 148: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/148.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 149: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/149.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 150: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/150.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 151: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/151.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 152: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/152.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 153: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/153.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 154: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/154.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 155: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/155.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .
![Page 156: Mathematical Billiards, Flat Surfaces and Dynamicspeople.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf · Mathematical Billiards A mathematical billiard consists of: a billiard](https://reader031.vdocuments.us/reader031/viewer/2022021717/5b3409a07f8b9aec518badc9/html5/thumbnails/156.jpg)
A ”simple” open question
The following question, very simple to formule, has been long OPEN andhas resisted many mathematicians’ efforts:
Take an acute triangle,there is a periodic trajectory(the Fagnano trajectory)
Take an obtuse triangle:is there a periodic trajectory?
Many questions about billiards with irrational angles are widely open . . .