Convex billiards on convex spheres

TL;DR

This paper explores the dynamics of convex billiards on convex spheres, showing generic properties of periodic points and establishing the nonlinear stability of elliptic points using Herman's Diophantine invariant curves result.

Contribution

It proves that generically, all periodic points are either hyperbolic or elliptic with irrational rotation, and demonstrates the nonlinear stability of elliptic points on convex spheres.

Findings

Periodic points are generically hyperbolic or elliptic with irrational rotation.

Hyperbolic periodic points have transverse homoclinic intersections.

Elliptic periodic points are nonlinearly stable for a dense set of convex billiards.

Abstract

In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that $C^\infty$ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is that we use Herman's result on Diophantine invariant curves to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.