Periodic Orbits of Oval Billiards on Surfaces of Constant Curvature

TL;DR

This paper investigates the dynamics of billiard systems on surfaces of constant curvature, establishing the genericity and density of nondegenerate periodic orbits, and analyzing their stability across different geometries.

Contribution

It introduces a new framework for billiards on curved surfaces, proving the openness and density of nondegenerate periodic orbits, and explores stability properties in these settings.

Findings

Finite nondegenerate periodic orbits form an open set.

Such orbits are dense in Euclidean and hyperbolic planes.

Spherical case differs due to perimeter constraints.

Abstract

In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary of the region is an oval. Using these properties and defining good perturbations for billiards, we show, in this new version, that having only a finite number of nondegenerate periodic orbits for each fixed period is an open property for billiards on surfaces of constant curvature and a dense one on the Euclidean and the hyperbolic planes. For the proof of the density, the techniques we use for the Euclidean and hyperbolic cases, do not work for the spherical case, due to a constraint (the perimeter of the polygonal trajectory being a multiple of {\pi}). We finish this paper studying the stability of these nondegenerate orbits.