Dissipative outer billiards: A case study

We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behaviour is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experiments suggesting that dissipative outer billiards about regular polygon are generically asymptotically periodic

The attractor of piecewise expanding maps of the interval

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e. the number of ergodic attractors and their corresponding mixing components do not change under small perturbations of the map. Our methods provide a topological description of the attractor and give an elementary proof of the density of periodic orbits

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai-Ruelle-Bowen measures whose basins cover a set of full Lebesgue measure

Ergodicity of polygonal slap maps

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 reflection laws. We study the absolutely continuous invariant probabilities (acips) of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps with more than one ergodic acip

Hyperbolic Polygonal Billiards Close to 1-Dimensional Piecewise Expanding Maps

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 have 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 probabilities (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is set equal to zero. We prove that if a convex polygon satisfies a generic condition called (*), and 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

Polygonal billiards with strongly contractive reflection laws: A review of some hyperbolic properties

We provide an overview of recent results concerning the dynamics of polygonal billiards with strongly contractive reflection laws

SRB Measures for Polygonal Billiards with Contracting Reflection Laws

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