SRB Measures for Polygonal Billiards with Contracting Reflection Laws

Abstract

Abstract. We prove that polygonal billiards with contracting reflection laws exhibit hyperbolic attractors with countably many ergodic SRB measures. These measures are robust under small perturbations of the reflection law, and the tables for which they exist form a generic set in the space of all convex polygons. Specific polygonal tables are studied in detail.

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

“…For billiards in generic convex polygons with a strong contracting reflection law, we have recently proved the existence of countably many ergodic Sinai-Ruelle-Bowen measures (SRB), each one supported on a uniformly hyperbolic attractor [9]. This result is significantly extended in the current paper by enlarging the class of allowed polygons, including now non-convex polygons, and more importantly by removing any restriction on the contraction factor of the reflection law (cf.…”

“…In this case, the billiard is a dissipative system: its map does not longer preserve an absolutely continuous measure, and may have attractors [1,2,8]. Indeed, if there are no period two orbits, then the map has a uniformly hyperbolic attractor [9]. Notice that period two orbits correspond to collisions perpendicular to a pair of parallel sides of the billiard table.…”

Ergodicity of polygonal slap maps

“…A detailed derivation of the reduced slap map can be found in [4,Section 6]. Clearly, when d is even, the slap map is an involution, i.e.…”

Microorganism billiards in closed plane curves

“…Current extensions to well-studied billiard problems include modifications of the table geometry, the shape of the inter-collision trajectories, and the rule for generating a new trajectory upon contact with the table boundary. In this direction, recent attention has been paid to aspecular reflection laws, especially in dissipative billiard systems commonly referred to as pinball billiards [8][9][10][11] and slap maps [12][13][14], as well as to aspecular reflection laws arising from other physical effects [15].…”