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Multi angle  On the ergodicity of billiards in non-rational polygons
Forni, Giovanni (Auteur de la Conférence) | CIRM (Editeur )

We will present a geometric criterion for the ergodicity of the billiard flow in a polygon with non-rational angles and discuss its application to the Diophantine case.

37D40 ; 37D50 ; 30F10 ; 30F60 ; 32G15

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Multi angle  On the algebraic hull of the Kontsevich-Zorich cocycle and applications to finiteness theorems
Eskin, Alex (Auteur de la Conférence) | CIRM (Editeur )

We give a necessary and sufficient condition for the existence of infinitely many non-arithmetic Teichmuller curves in a stratum of abelian differentials. This is joint work with Simion Filip and Alex Wright.

30F30 ; 32G15 ; 32G20 ; 14D07 ; 37D25

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t-align:justify"> We give an algebraic proof of the simplicity of the Lyapunov spectrum for the Teichmüller flow on strata of abelian differentials. This proof extends to the Kontsevich Zorich cocycle over strata of quadratic differentials and can also be used to study the algebraic degree of pseudo-Anosov stretch factors.

37D35

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Multi angle  Limits of geodesic push-forwards of horocycle measures
Forni, Giovanni (Auteur de la Conférence) | CIRM (Editeur )

We prove a couple of general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow. We prove a couple of general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of ...

37D40 ; 37C40 ; 37A17

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Multi angle  Interval exchange transformations from tiling billiards
Davis, Diana (Auteur de la Conférence) | CIRM (Editeur )

Tiling billiards is a dynamical system where beams of light refract through planar tilings. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, give related results, and show computer simulations of the system.

37D50 ; 37B50

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Multi angle  Unique ergodicity of geodesic flow in an infinite translation surface
Rafi, Kasra (Auteur de la Conférence) | CIRM (Editeur )

The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is uniquely ergodic in almost every direction. We call this class of surface essentially finite.
(joint work with Anja Randecker).
The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is ...

37D40 ; 51A40 ; 37A25

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Multi angle  Multiple mixing and Ratner property in area-preserving flows
Ulcigrai, Corinna (Auteur de la Conférence) | CIRM (Editeur )