# American Institute of Mathematical Sciences

March  2018, 38(3): 1145-1160. doi: 10.3934/dcds.2018048

## Uniform hyperbolicity in nonflat billiards

 Institut Fourier, Université Grenoble Alpes, 100, rue des mathématiques, 38610 Gières, France

Received  February 2017 Revised  July 2017 Published  December 2017

Fund Project: The author is supported by ERC advanced grant 320939

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian surface with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.

Citation: Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048
##### References:

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##### References:
The billiard reflection law
A grazing collision on a dispersing billiard in $\mathbb T^2$. The flow stops being defined after this time
On the left, a dispersing billiard in $\mathbb T^2$ with infinite horizon. On the right, a dispersing billiard in $\mathbb T^2$ with finite horizon
Each $A_k$ maps the cone $xy > 0$ (in grey) into the smaller cone $C_\epsilon$ (in dark grey)
$u$ is not well-defined at the convergence point
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