Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-25T00:39:32.765Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

Published online by Cambridge University Press:  14 March 2017

GIANLUIGI DEL MAGNO ,
JOÃO LOPES DIAS ,
PEDRO DUARTE  and
JOSÉ PEDRO GAIVÃO
GIANLUIGI DEL MAGNO
Affiliation:
Universidade Federal da Bahia, Instituto de Matemática, Avenida Adhemar de Barros, Ondina, 40170–110 Salvador, BA, Brasil email gdelmagno@ufba.br
JOÃO LOPES DIAS
Affiliation:
Departamento de Matemática and CEMAPRE, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200–781 Lisboa, Portugal email jldias@iseg.ulisboa.pt
PEDRO DUARTE
Affiliation:
Departamento de Matemática and CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749–016 Lisboa, Portugal email pmduarte@fc.ul.pt
JOSÉ PEDRO GAIVÃO
Affiliation:
Departamento de Matemática and CEMAPRE, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200–781 Lisboa, Portugal email jpgaivao@iseg.ulisboa.pt
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2062 - 2085
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arroyo, A., Markarian, R. and Sanders, D. P.. Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries. Nonlinearity 22(7) (2009), 1499–1522.Google Scholar
Arroyo, A., Markarian, R. and Sanders, D. P.. Structure and evolution of strange attractors in non-elastic triangular billiards. Chaos 22 (2012), 026107.Google Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.Google Scholar
Chernov, N.. Decay of correlations and dispersing billiards. J. Stat. Phys. 94(3–4) (1999), 513–556.Google Scholar
Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127) . American Mathematical Society, Providence, RI, 2006.Google Scholar
Chernov, N. and Zhang, H.-K.. Billiards with polynomial mixing rates. Nonlinearity 18(4) (2005), 1527–1553.Google Scholar
Chernov, N. and Zhang, H.-K.. On statistical properties of hyperbolic systems with singularities. J. Stat. Phys. 136(4) (2009), 615–642.Google Scholar
Del Magno, G., Lopes Dias, J., Duarte, P., Gaivão, J. P. and Pinheiro, D.. Chaos in the square billiard with a modified reflection law. Chaos 22 (2012), 026106.Google Scholar
Del Magno, G., Lopes Dias, J., Duarte, P., Gaivão, J. P. and Pinheiro, D.. SRB measures for polygonal billiards with contracting reflection laws. Comm. Math. Phys. 329 (2014), 687–723.Google Scholar
Del Magno, G., Lopes Dias, J., Duarte, P. and Gaivão, J. P.. Ergodicity of polygonal slap maps. Nonlinearity 27(8) (2014), 1969–1983.Google Scholar
Del Magno, G., Lopes Dias, J., Duarte, P. and Gaivão, J. P.. Ergodic properties of polygonal billiards with strongly contracting reflection laws, Preprint, 2015, arXiv:1501.03697.Google Scholar
Erickson, L. H. and La Valle, S. M.. Toward the design and analysis of blind, bouncing robots. IEEE Int. Conf. on Robotics and Automation (Karlsruhe, Germany, 6–10 May 2013). IEEE, Piscataway, NJ, 2013.Google Scholar
Katok, A. et al. . Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222) . Springer, Berlin, 1986.Google Scholar
Markarian, R., Pujals, E. J. and Sambarino, M.. Pinball billiards with dominated splitting. Ergod. Th. & Dynam. Sys. 30(6) (2010), 1757–1786.Google Scholar
Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 339–351.Google Scholar
Pesin, Y. B.. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. & Dynam. Sys. 12(1) (1992), 123–151.Google Scholar
Sataev, E. A.. Invariant measures for hyperbolic mappings with singularities. Uspekhi Mat. Nauk 47(1(283)) (1992), 147–202, 240; Engl. trans. Russian Math. Surveys 47(1) (1992), 191–251.Google Scholar
Sheriff, R. E. and Geldart, L. P.. Exploration Seismology. Cambridge University Press, Cambridge, 1995.Google Scholar