Expansiveness and hyperbolicity in convex billiards

Bessa, Mário; Dias, João Lopes; Torres, Maria Joana

http://hdl.handle.net/10400.2/13899

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Authors

Bessa, Mário

Dias, João Lopes

Torres, Maria Joana

Abstract(s)

We say that a convex planar billiard table B is C2-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 Λ_{B,U}, and this property holds under C2-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 Λ_{B,U} is uniformly hyperbolic. In addition, we show that this property also holds for a generic choice among billiards which are expansive.