Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-16T23:23:52.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Scaling rate for semi-dispersing billiards with non-compact cusps

Published online by Cambridge University Press:  08 November 2011

A. ARBIETO ,
R. MARKARIAN ,
M. J. PACIFICO  and
R. SOARES
A. ARBIETO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil (email: arbieto@im.ufrj.br, pacifico@im.ufrj.br)
R. MARKARIAN
Affiliation:
Instituto de Matemática y Estadística (IMERL), Facultad de Ingeniería, Universidad de la República, CC30, CP 11300, Montevideo, Uruguay (email: roma@fing.edu.uy)
M. J. PACIFICO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil (email: arbieto@im.ufrj.br, pacifico@im.ufrj.br)
R. SOARES
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil (email: regis.soares@ufjf.edu.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that certain billiard tables with non-compact cusps are mixing with respect to the invariant infinite measure, in the sense of Krengel and Sucheston. Moreover, we show that the scaling rate is slower than a certain polynomial rate.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 1836 - 1861
Copyright
Copyright © Cambridge University Press 2011

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

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.Google Scholar
[2]Bunimovich, L. A. and Sinai, Ya. G.. Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 (1981), 479–497.Google Scholar
[3]Chernov, N. I.. Decay of correlations and dispersing billiards. J. Stat. Phys. 94(3–4) (1999), 513–556.Google Scholar
[4]Chernov, N. and Markarian, R.. Introduction to the Ergodic Theory of Chaotic Billiards (24o Colóquio Brasileiro de Matemática). Publicações Matemáticas, IMPA, Rio de Janeiro, 2003.Google Scholar
[5]Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127). American Mathematical Society, Providence, RI, 2006.Google Scholar
[6]Chernov, N. and Markarian, R.. Dispersing billiards with cusps: slow decay of correlations. Comm. Math. Phys. 270(3) (2007), 727–758.CrossRefGoogle Scholar
[7]Chernov, N. and Zhang, H.-K.. Improved estimates for correlations in billiards. Comm. Math. Phys. 277(2) (2008), 305–321.CrossRefGoogle Scholar
[8]Conway, J. B.. A Course in Functional Analysis, 2nd edn(Graduate Texts in Mathematics, 96). Springer, New York, 1990.Google Scholar
[9]Coudene, Y.. On invariant distributions and mixing. Ergod. Th. & Dynam. Sys. 27 (2007), 109–112.CrossRefGoogle Scholar
[10]Danilenko, A. I. and Silva, C. E.. Ergodic Theory: Non-singular Transformations (Encyclopedia of Complexity and Systems Science). Springer, 2009, pp. 3055–3083.Google Scholar
[11]Isola, S.. Renewal sequences and intermittency. J. Stat. Phys. 97(1–2) (1999), 263–280.CrossRefGoogle Scholar
[12]Isola, S.. On systems with finite ergodic theory. Far East J. Dyn. Syst. 5(1) (2003), 1–62.Google Scholar
[13]Katok, A. and Strelcyn, J.-M.. (in collaboration with F. Ledrappier and F. Przytycki). Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Spinger, 1986.CrossRefGoogle Scholar
[14]Krengel, U. and Sucheston, L.. On mixing in infinite measure spaces. Z. Wah. Verw. Geb. 13 (1969), 150–164.CrossRefGoogle Scholar
[15]Lenci, M.. Semi-dispersing billiards with an infinite cusp I. Comm. Math. Phys. 230(1) (2002), 133–180.Google Scholar
[16]Lenci, M.. On infinite-volume mixing. Comm. Math. Phys. 298(2) (2010), 485–514.CrossRefGoogle Scholar
[17]Liverani, C. and Wojtkowski, M.. Ergodicity in Hamiltonian systems. Dynamics Reported (Dynamics Reported, Expositions Dynam. Systems (N.S.), 4). Springer, Berlin, 1995, pp. 130–202.Google Scholar
[18]Markarian, R.. Billiards with polynomial decay of correlations. Ergod. Th. & Dynam. Sys. 24 (2004), 177–197.Google Scholar
[19]Riesz, F. and Nagy, B. Sz.. Functional Analysis. Ed. Boron, L. F.. Dover Publications, Inc., New York, 1990, Translated from the 2nd french edition.Google Scholar
[20]Rudin, W.. Real and Complex Analysis, 3rd edn. WCB/McGraw-Hill, New York, 1987.Google Scholar
[21]Sinai, Ya. G.. Dynamical systems with elastic reflections. Russian Math. Surveys 25 (1970), 137–189.Google Scholar
[22]Sucheston, L.. On mixing and the zero-one law. J. Math. Anal. Appl. 6 (1963), 447–456.Google Scholar
[23]Wojtkowski, M.. Principles for the design of billiards with non-vanishing Lyapunov exponents. Comm. Math. Phys. 105 (1986), 391–414.CrossRefGoogle Scholar
[24]Young, L. S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.CrossRefGoogle Scholar
[25]Young, L. S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar