Expansiveness and Hyperbolicity in Convex Billiards

    2021, Volume 26, Number 6, pp.  756-762

    Author(s): Bessa M., Lopes-Dias J., Torres M.

    We say that a convex planar billiard table $B$ is $C^2$-stably expansive on a fixed open subset $U$ of the phase space if its billiard map $f_B$ is expansive on the maximal invariant set $\Lambda_{B,U}=\bigcap_{n\in\mathbb{Z}}f^n_B(U)$, and this property holds under $C^2$-perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of $f_B$ in $\Lambda_{B,U}$ is uniformly hyperbolic. In addition, we show that this property also holds for a generic choice among billiards which are expansive.
    Keywords: convex planar billiards, hyperbolic sets, expansiveness
    Citation: Bessa M., Lopes-Dias J., Torres M., Expansiveness and Hyperbolicity in Convex Billiards, Regular and Chaotic Dynamics, 2021, Volume 26, Number 6, pp. 756-762

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