Dinâmica genérica de bilhares ovais em superfícies de curvatura constante: estendendo alguns resultados do plano

Abstract

We use the C2 topology to investigate generic proprieties for oval billiards on the plane, sphere and hyperbolic plane. Together with the work by dos Santos and Pinto de Carvalho (2017) , we extend the results of Dias Carneiro et al.(2007) about plane oval billiards to oval billiards on the sphere and the hyperbolic plane. We are going to show that, under certain generic conditions, oval billiards on these surfaces have only a finite number of periodic orbits, for each period N, all nondegenerate. Moreover, the stable and unstable manifolds of two hyperbolic points either do not intersect or have at least one transversal intersection. We also show that any oval billiard with a 2-periodic elliptic orbit can be approximated by a billiard with elliptic islands, by calculating the First Birkhoff Coefficient.