Dois resultados em bilhares em superfícies com curvatura constante

Abstract

In this work we extend some results about plane billiards to the hyperbolic plane and to ahemisphere of the sphere. First we consider billiards defined in the region bounded by a closed and geodesically strictly convex curve. Studying the polynomial entropy of these billiards we prove that the circular billiard map has polynomial entropy equal to 1, while other billiards have polynomial entropy >=2. In particular, we prove that the elliptical billiard has polynomial entropy equal to 2. The main tools used were the integrability of circular and elliptical billiards on surfaces with constant curvature, the Twist property of those billiard maps and a generalization of the techniques applied by Marco [27] to calculate polynomial entropy. In the second part of the work, we consider stadium-like billiard tables and show that, when the focusing parts are connected by sufficiently long geodesic segments, the billiard map has a positive Lyapunov exponent almost everywhere. The main tools used were a Wojtkowski's version [40] of the cone field method and a generalization of the construction of the cone fields presented by Donnay [14]. We finish this part by studying the circular stadium billiard in the hyperbolic plane.