Expansiveness and Hyperbolicity in Convex Billiards

Abstract

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.