Ergodicity of polygonal slap maps

Abstract

Abstract. Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.

Hyperbolic billiards on polytopes with contracting reflection laws

“…Hence the escaping time is bounded on the chamber. This concept generalizes the notion of chamber introduced in [6].…”

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

“…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. [8,9,10,11]). In addition, we establish that the basins of the ergodic SRB measures cover a set of full Lebesgue measure.…”

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].…”