# 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.

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