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- Expansiveness and hyperbolicity in convex billiardsPublication . Bessa, Mário; Dias, João Lopes; Torres, Maria JoanaWe 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.
- Hyperbolicity through stable shadowing for generic geodesic flowsPublication . Bessa, Mário; Dias, João Lopes; Torres, Maria JoanaWe prove that the closure of the closed orbits of a generic geodesic flow on a closed Riemannian n ≥ 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds C2-robustly on the metric. We obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization.
- On shadowing and hyperbolicity for geodesic flows on surfacesPublication . Bessa, Mário; Dias, João Lopes; Torres, Maria JoanaWe prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in Bessa et al. (2013) for Hamiltonian systems.