Breaking of invariant curves: from the Fermi-Ulam map to the breathing circle billiard

TL;DR

This paper analyzes the dynamics of a periodically breathing circle billiard, showing how invariant curves break down at small angular momentum, leading to positive topological entropy and chaotic behavior.

Contribution

It provides a sharper quantitative threshold for the destruction of invariant curves in the breathing circle billiard, extending the analysis from the Fermi-Ulam map.

Findings

Invariant Lipschitz graphs are excluded for certain rotation numbers.

Positive topological entropy is established for small angular momentum.

The results improve previous thresholds for invariant curve destruction.

Abstract

We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and fixing its value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder. In the high-energy regime this map is a twist map generated by a diagonally periodic generating function $h_c$. We study the small angular momentum regime as a perturbation of the limiting case $c=0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather's converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni's theorem, this yields positive topological entropy for sufficiently small $c\geq0$. Our result improves previous similar results obtained via the standard Mather's converse-KAM criterion by giving a sharper quantitative threshold for the destruction of invariant curves.