Expansiveness and Hyperbolicity in Convex Billiards

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.