Billiard dynamics near an invariant horizontal circle

TL;DR

This paper investigates the dynamics near an invariant horizontal circle in certain oval billiards, revealing how other invariant curves approach it and discussing the resulting dynamical implications.

Contribution

It characterizes the behavior of invariant rotational curves near the horizontal circle in constant width and symmetric oval billiards, a novel analysis in billiard dynamics.

Findings

Invariant horizontal circle is approached by other invariant curves from both sides.

Dynamics near the circle exhibit specific convergence properties.

Describes consequences for the overall billiard system behavior.

Abstract

Sufficiently differentiable oval billiards always have invariant rotational curves, but there are only two types of ovals with an invariant horizontal circle in its phase-space: the constant width ovals and some very special symmetric curves. In this work we study the dynamics near the horizontal circle for the billiard map of those two types of ovals, and show that the horizontal circle is approached, from both sides, by other invariant rotational curves. We will also describe some dynamical consequences.