Hyperbolic polygonal billiards close to 1-dimensional piecewise expanding maps

TL;DR

This paper explores the relationship between hyperbolic polygonal billiards with contracting reflection laws and one-dimensional piecewise expanding maps, establishing a correspondence between their ergodic measures under certain conditions.

Contribution

It demonstrates a one-to-one correspondence between ergodic SRB measures of the billiard map and ergodic acips of the slap map for generic polygons with small Lipschitz constants.

Findings

Established a correspondence between billiard and slap map measures.

Analyzed the case of regular polygons and triangles.

Connected the number of Bernoulli components to ergodic acips.

Abstract

We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards has a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probability measures (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is always equal to zero. Our main result states that for a generic polygon, if the reflection law has a Lipschitz constant sufficiently small, then there exists a one-to-one correspondence between the ergodic SRB measures of the billiard map and the ergodic acips of the corresponding slap map, and moreover that the number of Bernoulli components of each ergodic SRB measure equals the number of the exact components of the corresponding ergodic acip. The case of billiards in regular polygons and triangles is studied in detail.