Hyperbolic billiards with nearly flat focusing boundaries. I

TL;DR

This paper constructs hyperbolic billiards with nearly flat focusing boundaries, challenging the traditional requirement that the table's diameter be proportional to the boundary's curvature, using a novel cone bundle approach.

Contribution

It introduces a new method to prove hyperbolicity in billiards with very flat focusing boundaries, bypassing the standard defocusing mechanism.

Findings

Hyperbolic billiards with arbitrarily small boundary curvature are possible.

The diameter of these billiards can be bounded independently of boundary curvature.

A new cone bundle technique is developed for the proof.

Abstract

The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstardard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms.