Bilhares com obstáculos

Abstract

We consider planar focusing billiards on annular tables constructed by adding a circular obstacle of center p and radius r in the interior of a region bounded by a convex curve . Parameterized families of billiards are obtained by fixing the pair (...) and by taking the radius r as a parameter. We show that for a generically choice of (...) and for r small enough, the map induced by the return of trajectories to the obstacle admits uniformly hyperbolic sets in neighborhoods of periodic orbits whose corresponding trajectories have perpendicular collisions with the obstacle. Generic conditions for the existence of such orbits are given and the geometry of the corresponding hyperbolic sets is described. In the case that the external boundary is a unitary circle and the center of obstacle has distance (...) from the center of , we consider a two parameter (...) family of billiards to obtain hyperbolic sets (...) which are (...) dense in phase space with (...) as (...). We also show the existence of parameters, arbitrarily close to (1,0), such that the corresponding billiard admits homoclinic tangences and linear elliptic periodic orbits with are (...) dense in phase space.