Chaoticity of generic analytic convex billiards

TL;DR

This paper proves that generic analytic strongly convex billiards exhibit maximal chaos, with all intersections of stable and unstable manifolds of periodic orbits being transverse for all rational rotation numbers.

Contribution

It establishes the generic chaotic behavior of analytic convex billiards by demonstrating transversality of manifold intersections for all rational rotation numbers.

Findings

All intersections between stable and unstable manifolds are transverse.

The chaotic behavior applies to all rational rotation numbers.

The result holds for generic analytic strongly convex billiards.

Abstract

We show that a generic analytic strongly convex billiard is "maximally chaotic" in the sense that, for every rational number $\frac{p}{q} \in \mathbb{Q} \cap (0,1)$, all intersections between the stable and unstable manifolds of maximizing periodic orbits with rotation number $\frac{p}{q}$ are transverse.