Uniform hyperbolicity in nonflat billiards

Mickaël Kourganoff

doi:10.3934/dcds.2018048

Article Contents

2018, Volume 38, Issue 3: 1145-1160. Doi: 10.3934/dcds.2018048

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Uniform hyperbolicity in nonflat billiards

Mickaël Kourganoff

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

Received: January 31, 2017

Revised: June 30, 2017

Published: June 2018

The author is supported by ERC advanced grant 320939

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Abstract

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.

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Mathematics Subject Classification: 37D20, 53D25, 70E55.

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Figure 1. The billiard reflection law

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Figure 2. A grazing collision on a dispersing billiard in $\mathbb T^2$. The flow stops being defined after this time

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Figure 3. 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

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Figure 4. Each $A_k$ maps the cone $xy > 0$ (in grey) into the smaller cone $C_\epsilon$ (in dark grey)

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Figure 5. $u$ is not well-defined at the convergence point

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