Billiards in generic convex bodies have positive topological entropy

TL;DR

This paper proves that in most smooth convex bodies, billiard dynamics are chaotic with positive topological entropy, indicating complex and unpredictable behavior.

Contribution

It establishes that generic smooth convex bodies have billiard maps with hyperbolic sets, leading to chaos and exponential orbit growth.

Findings

Existence of hyperbolic basic sets in generic convex billiards

Positive topological entropy for billiards in generic convex bodies

Exponential growth in the number of periodic orbits

Abstract

We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.