Existence and Nonexistence of Invariant Curves of Coin Billiards

TL;DR

This paper investigates the existence of invariant curves in coin billiards, proving their presence in certain scenarios and absence in others, with special focus on the circular coin case and numerical experiments on elliptical coins.

Contribution

It establishes conditions for the existence and nonexistence of invariant curves in coin billiards, extending Bialy's questions and analyzing specific geometric configurations.

Findings

Existence of KAM curves near boundary for small or near-circular coins.

Absence of invariant curves for large noncircular coins.

Circular coin billiard uniquely foliated by invariant curves.

Abstract

In this paper we consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map $T$ of the annulus $\mathbb A = \mathbb T \times (0,\pi)$. We prove the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary $\partial \mathbb A$; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of $\partial \mathbb A$ through which there passes no invariant essential curve; and the only coin billiard for which the phase space $\mathbb A$ is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, we describe the results of some numerical experiments on the elliptical coin billiard.