Hyperbolicity and Abundance of Elliptical Islands in Annular Billiards

TL;DR

This paper investigates the complex dynamics of billiards in annular tables with excentric circles, revealing hyperbolic sets and elliptical islands that coexist, influenced by parameter changes.

Contribution

It introduces a two-parameter family of billiard maps showing hyperbolic sets and elliptical islands, demonstrating coexistence phenomena in annular billiards.

Findings

Existence of hyperbolic sets increasing with parameter changes

Presence of quadratic homoclinic tangencies leading to elliptical islands

Coexistence of hyperbolic dynamics with elliptical islands

Abstract

We study the billiard dynamics in annular tables between two excentric circles. As the center and the radius of the inner circle change, a two parameters map is defined by the first return of trajectories to the obstacle. We obtain an increasing family of hyperbolic sets, in the sense of the Hausdorff distance, as the radius goes to zero and the center of the obstacle approximates the outer boundary. The dynamics on each of these sets is conjugate to a shift with an increasing number of symbols. We also show that for many parameters the system presents quadratic homoclinic tangencies whose bifurcation gives rise to elliptical islands (Conservative Newhouse Phenomenon). Thus, for many parameters we obtain the coexistence of a "large" hyperbolic set with many elliptical islands.