Hyperbolic billiards on polytopes with contracting reflection laws

TL;DR

This paper investigates hyperbolic dynamics of billiards in polytopes with contracting reflection laws, establishing conditions for uniform hyperbolicity and analyzing specific cases like 3D simplexes.

Contribution

It introduces new conditions under which billiards with contracting laws are uniformly hyperbolic, extending understanding of dynamical behavior in non-standard billiard systems.

Findings

Billiards on generic polytopes are uniformly hyperbolic under certain geometric conditions.

Uniform hyperbolicity holds when the reflection law is close to specular or the polytope is obtuse.

Detailed analysis of billiards on a family of 3D simplexes.

Abstract

We study billiards on polytopes in $\Rr^d$ with contracting reflection laws, i.e. non-standard reflection laws that contract the reflection angle towards the normal. We prove that billiards on generic polytopes are uniformly hyperbolic provided there exists a positive integer $k$ such that for any $k$ consecutive collisions, the corresponding normals of the faces of the polytope where the collisions took place generate $\Rr^d$. As an application of our main result we prove that billiards on generic polytopes are uniformly hyperbolic if either the contracting reflection law is sufficiently close to the specular or the polytope is obtuse. Finally, we study in detail the billiard on a family of $3$-dimensional simplexes.