Existence and Nonexistence of Invariant Curves of Coin Billiards

TL;DR

This work analyzes the coin billiard map T on the annulus A = T × (0, π), arising from a nonsmooth geodesic flow on a cylinder with top and bottom copies of a convex table. The authors develop a generating function framework and apply KAM theory to prove the existence of invariant curves in two perturbative regimes (near-circular tables and small coin height), while also proving nonexistence of near-boundary invariant curves for large height when the table is not circular, and establishing that only the circular disk yields a phase space foliated by essential invariant curves. They provide two distinct proofs for nonexistence near the boundary, leveraging variational (Mather) methods and Lipschitz/Herman bounds, and extend a theorem of Bialy to classify integrability versus nonintegrability in coin billiards. The results yield a near-complete picture: coin billiards are not generically integrable, with invariant curves persisting only in highly constrained perturbative regimes, and numerical experiments corroborate the theoretical findings, illustrating the transition from regular to chaotic dynamics as table geometry or height varies. Overall, the work advances understanding of invariant structures in nonsmooth billiard-type systems and connects to classical results on twist maps and integrability.

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 $\mathbb A = \mathbb T \times (0,π)$. 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.

Figures (4)

• Figure 1: A schematic picture of the motion of the particle on the configuration coin.
• Figure 2: A depiction of the proof of Theorem \ref{['theorem_birkhoffzones']}.
• Figure 3: Each picture shows the first 100 iterations of various initial conditions under the coin mapping $T$ corresponding to the coin $\mathcal{C} \left( \mathcal{E}_{a,b},\ell \right)$, where $\mathcal{E}_{a,b}$ is ellipse with semimajor axis $a$ and semiminor axis $b$. We chose 144 different initial conditions spread evenly through the phase space. Different colours represent different orbits. The horizontal axis is the arclength parameter, and the vertical axis is the angle of incidence/reflection.
• Figure 4: Each picture shows the first 100 iterations of various initial conditions under the coin mapping $T$ corresponding to the coin $\mathcal{C} \left( \mathcal{E}_{a,b},\ell \right)$, where $\mathcal{E}_{a,b}$ is ellipse with semimajor axis $a$ and semiminor axis $b$. We chose 144 different initial conditions spread evenly through the phase space. Different colours represent different orbits. The horizontal axis is the arclength parameter, and the vertical axis is the angle of incidence/reflection.

Theorems & Definitions (33)

• Theorem A
• Theorem B
• Theorem C
• Corollary D
• Corollary E
• Corollary F
• Example
• Lemma 1
• proof
• Theorem 2