June 2017, 37(6): 3079-3109. doi: 10.3934/dcds.2017132

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  February 2017

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.

Citation: Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132
References:
[1]

A. ArroyoR. Markarian and D. P. Sanders, Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries, Nonlinearity, 22 (2009), 1499-1522. doi: 10.1088/0951-7715/22/7/001.

[2]

A. Arroyo, R. Markarian and D. P. Sanders, Structure and evolution of strange attractors in non-elastic triangular billiards Chaos 22 (2012), 026107, 12pp. doi: 10.1063/1.4719149.

[3]

P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles; Continuity Via Large Deviations Atlantis Studies in Dynamical Systems, vol. 3, Atlantis Press, 2016. doi: 10.2991/978-94-6239-124-6.

[4]

G. Del Magno, J. Lopes Dias, P. Duarte, J. P. Gaivão and D. Pinheiro, Chaos in the square billiard with a modified reflection law Chaos 22 (2012), 026106, 11pp. doi: 10.1063/1.3701992.

[5]

G. Del MagnoJ. Lopes DiasP. DuarteJ. P. Gaivão and D. Pinheiro, SRB measures for polygonal billiards with contracting reflection laws, Comm. Math. Phys., 329 (2014), 687-723. doi: 10.1007/s00220-014-1960-x.

[6]

G. Del MagnoJ. Lopes DiasP. Duarte and J. P. Gaivão, Ergodicity of polygonal slap maps, Nonlinearity, 27 (2014), 1969-1983. doi: 10.1088/0951-7715/27/8/1969.

[7]

G. Del Magno, J. Lopes Dias, P. Duarte and J. P. Gaivão, Hyperbolic polygonal billiards with finitely may ergodic SRB measures, to appear in Ergodic Theory Dyn. Syst. (2016), arXiv:1507.06250.

[8]

R. MarkarianE. R. Pujals and M. Sambarino, Pinball billiards with dominated splitting, Ergodic Theory Dyn. Syst., 30 (2010), 1757-1786. doi: 10.1017/S0143385709000819.

[9]

Ya. G. Sinai, Billiard trajectories in a polyhedral angle, Russian Math. Surveys, 33 (1978), 229-230.

[10]

S. Sternberg, Lectures on Differential Geometry Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1964.

show all references

References:
[1]

A. ArroyoR. Markarian and D. P. Sanders, Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries, Nonlinearity, 22 (2009), 1499-1522. doi: 10.1088/0951-7715/22/7/001.

[2]

A. Arroyo, R. Markarian and D. P. Sanders, Structure and evolution of strange attractors in non-elastic triangular billiards Chaos 22 (2012), 026107, 12pp. doi: 10.1063/1.4719149.

[3]

P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles; Continuity Via Large Deviations Atlantis Studies in Dynamical Systems, vol. 3, Atlantis Press, 2016. doi: 10.2991/978-94-6239-124-6.

[4]

G. Del Magno, J. Lopes Dias, P. Duarte, J. P. Gaivão and D. Pinheiro, Chaos in the square billiard with a modified reflection law Chaos 22 (2012), 026106, 11pp. doi: 10.1063/1.3701992.

[5]

G. Del MagnoJ. Lopes DiasP. DuarteJ. P. Gaivão and D. Pinheiro, SRB measures for polygonal billiards with contracting reflection laws, Comm. Math. Phys., 329 (2014), 687-723. doi: 10.1007/s00220-014-1960-x.

[6]

G. Del MagnoJ. Lopes DiasP. Duarte and J. P. Gaivão, Ergodicity of polygonal slap maps, Nonlinearity, 27 (2014), 1969-1983. doi: 10.1088/0951-7715/27/8/1969.

[7]

G. Del Magno, J. Lopes Dias, P. Duarte and J. P. Gaivão, Hyperbolic polygonal billiards with finitely may ergodic SRB measures, to appear in Ergodic Theory Dyn. Syst. (2016), arXiv:1507.06250.

[8]

R. MarkarianE. R. Pujals and M. Sambarino, Pinball billiards with dominated splitting, Ergodic Theory Dyn. Syst., 30 (2010), 1757-1786. doi: 10.1017/S0143385709000819.

[9]

Ya. G. Sinai, Billiard trajectories in a polyhedral angle, Russian Math. Surveys, 33 (1978), 229-230.

[10]

S. Sternberg, Lectures on Differential Geometry Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1964.

Figure 1.  Barycentric angle $\phi$.
Figure 2.  Composition of the projections $P_{{v'}^ \perp}\circ P_{v,\eta^ \perp}$
Figure 3.  Parameter regions with uniform bounded escaping time
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