Elliptic islands and zero measure escaping orbits in a class of outer billiards

TL;DR

This paper investigates outer billiard systems around circular sectors, demonstrating the coexistence of stable elliptic islands and zero measure escaping orbits, revealing complex dynamical behaviors.

Contribution

It proves the existence of positive measure elliptic islands in semi-disc billiards and identifies sectors with zero measure escaping orbits, advancing understanding of system stability and diffusion.

Findings

Elliptic islands occupy a positive proportion of the plane in semi-disc billiards.

Certain circular sectors have zero measure of escaping orbits.

The coexistence of stability and diffusion is established in these systems.

Abstract

We study outer billiard systems around a class of circular sectors. For semi-discs, we prove the existence of elliptic islands occupying a positive proportion of the plane. Combined with known results, this shows the coexistence of stability and diffusion for this system. On the other hand, we show that there exists a countable family of circular sectors for which the outer billiard system has zero measure of escaping orbits.