Generic Oval Billiards

TL;DR

This paper proves that generic oval billiards have finitely many non-degenerate periodic orbits per period, with at least one hyperbolic orbit, and explores implications for their dynamical behavior, especially in instability regions.

Contribution

It establishes generic conditions ensuring finite periodic orbits and hyperbolicity in oval billiards, advancing understanding of their complex dynamics.

Findings

Finitely many periodic orbits per period under generic conditions

Existence of at least one hyperbolic periodic orbit per period

Transversal invariant curves between hyperbolic points

Abstract

In this paper we show that, under certain generic conditions, billiards on ovals have only a finite number of periodic orbits, for each period, all non-degenerate and at least one of them is hyperbolic. Moreover, the invariant curves of two hyperbolic points are transversal. We explore these properties to give some dynamical consequences specially about the dynamics in the instability regions.