Hyperbolic billiards on polytopes with contracting reflection laws

Pedro Duarte; José Pedro GaivÃo; Mohammad Soufi

doi:10.3934/dcds.2017132

Article Contents

2017, Volume 37, Issue 6: 3079-3109. Doi: 10.3934/dcds.2017132

This issue Previous Article Onofri inequalities and rigidity results Next Article On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations

Hyperbolic billiards on polytopes with contracting reflection laws

1.
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
2.
Departamento de Matemática and CEMAPRE, ISEG, Universidade de Lisboa, Rua do Quelhas 6,1200-781 Lisboa, Portugal
3.
Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, R. São Francisco Xavier 524,20550-900 Rio de Janeiro -RJ, Brasil

Received: February 2016

Revised: January 2017

Published: May 2017

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Abstract

We study billiards on polytopes in ${\mathbb{R}^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 ${\mathbb{R}^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.

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Mathematics Subject Classification: Primary: 37D50, 37D20; Secondary: 37D45.

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Figure 1. Barycentric angle $\phi$.

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Figure 2. Composition of the projections $P_{{v'}^ \perp}\circ P_{v,\eta^ \perp}$

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Figure 3. Parameter regions with uniform bounded escaping time

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Figure 4.

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