Chaotic motion in the breathing circle billiard

TL;DR

This paper demonstrates that even with regular boundary motion, a particle inside a circular billiard can exhibit chaotic behavior, with invariant measures of positive entropy, using variational methods.

Contribution

It introduces a class of boundary motions in circular billiards that lead to chaotic dynamics, extending understanding of chaos in time-dependent billiard systems.

Findings

Existence of invariant measures with positive entropy

Chaotic motion possible under regular boundary conditions

Application of Aubry-Mather theory to billiard dynamics

Abstract

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory.