On another edge of defocusing: hyperbolicity of asymmetric lemon billiards

TL;DR

This paper proves that asymmetric lemon billiards, with two close circular arcs, exhibit hyperbolic behavior, expanding understanding of chaotic billiard systems with focusing components.

Contribution

It demonstrates hyperbolicity in a new class of convex billiards called asymmetric lemons, even with closely situated focusing arcs, challenging previous separation conditions.

Findings

Asymmetric lemon billiards are hyperbolic despite close focusing arcs.

The result applies to perturbations of classical convex ergodic billiards.

Hyperbolicity is established for a novel class of billiard tables.

Abstract

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables--asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than forty years ago.