Bilhares convexos em superfícies de curvatura constante

Abstract

We consider a simple closed and geodesically strictly convex curve on hemisphere or hyperbolic plane and a moving particle free geodesic within the region bounded by this curve suffering elastic collisions with the curve at the points of shock. We show that the billiard map on these curves in these surfaces are a conservative difieomorphism twist-like,we will establish sucient conditions for non-persistence of resonant curves in perturbed gedesic circular billiards. We also show that billiards in these regions generically have a nite number of periodic orbits of any period n and they are all hyperbolic. We also established that the set of orbits of period three has Hausdorff dimension between zero and one, and in the latter case the tangent line at almost every point