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Uniform hyperbolicity in nonflat billiards

The author is supported by ERC advanced grant 320939

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  • 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.

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


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  • Figure 1.  The billiard reflection law

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

    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

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

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

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