Skip to main content
SpringerLink
Log in

Dispersing Billiards with Small Holes

  • Conference paper
  • First Online:
Ergodic Theory, Open Dynamics, and Coherent Structures

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 70))

Abstract

We study several classes of dispersing billiards with holes, including both finite and infinite horizon Lorentz gases and tables with corner points. We allow holes in the form of arcs in the boundary and open sets in the interior of the table as well as generalized holes in which escape may depend on the angle of collision as well as the position. For a large class of initial distributions (including Lebesgue measure and the smooth invariant (SRB) measure for the billiard before the introduction of the hole), we prove the existence of a common escape rate and a limiting conditionally invariant distribution. The limiting distribution converges to the SRB measure for the billiard as the hole tends to zero. Finally, we are able to characterize the common escape rate via pressure on the survivor set.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We give the definition for invertible T. When T is not invertible, define \({\mathring{M}}^{\infty } = \cap _{i=0}^{\infty }{T}^{-i}(M\setminus H)\).

  2. 2.

    In the presence of cusps (corner points whose angle is zero), it was proved in [CM07, CZ08] that the billiard map has polynomial decay of correlations and so in general will have polynomial rates of escape. Thus the present methods will not apply.

  3. 3.

    According to [23, Sect. 2.1], we could take \(m_{W}(N_{\varepsilon }(\partial H) \cap W) \leq C{_{0}\varepsilon }^{t_{0}}\) for any exponent t 0 > 0. We choose \(\varepsilon _{0} = 1/2\) here to simplify the exposition and choice of constants in the norms and because this already contains an interesting class of examples (see also (A3)(2) of Sect. 8.3.3).

  4. 4.

    Here by \(\tilde{{\mathcal{C}}}^{1}({\mathcal{W}}^{s})\), we mean to indicate \(\tilde{{\mathcal{C}}}^{p}({\mathcal{W}}^{s})\) with p = 1, i.e., functions which are Lipschitz on elements of \({\mathcal{W}}^{s}\).

  5. 5.

    Our treatment of stable curves here differs from that in [23]. In that abstract setting, stable curves are defined via graphs in charts of the given manifold. In the present more concrete setting, we dispense with charts and use the global \((r,\varphi )\) coordinates.

  6. 6.

    Recall that a physical measure for T is an ergodic, invariant probability measure μ for which there exists a positive Lebesgue measure set B μ , with μ(B μ ) = 1, such that \(\lim _{n\rightarrow \infty }\frac{1} {n}\sum _{i=0}^{n-1}\psi ({T}^{i}x) =\mu (\psi )\) for all x ∈ B μ and all continuous functions ψ.

  7. 7.

    Indeed, [11, Sect. 4.9] does not address the infinite horizon case explicitly, but a quick calculation shows that an exponent of 1∕3 is in fact needed.

References

  1. Baladi, V.: Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol.16, World Scientific, Singapore (2000)

    Google Scholar 

  2. Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotonic transformations. Comm. Math. Phys. 127, 459–477 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bruin, H., Demers, M., Melbourne, I.: Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Th. Dynam. Sys. 30, 687–728 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bunimovich, L., Sinai, Ya. G., Chernov, N.: Markov partitions for two-dimensional hyperbolic billiards. Russian Math. Surveys 45, 105–152 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5.  Bunimovich, L., Sinai, Ya. G., Chernov, N.: Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surv. 46, 47–106 (1991)

    Google Scholar 

  6. Čencova, N.N.: A natural invariant measure on Smale’s horseshoe. Soviet Math. Dokl. 23, 87–91 (1981)

    Google Scholar 

  7. Čencova, N.N.: Statistical properties of smooth Smale horseshoes. In: Dobrushin, R.L. (ed.) Mathematical Problems of Statistical Mechanics and Dynamics, pp. 199–256. Reidel, Dordrecht (1986)

    Chapter  Google Scholar 

  8. Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chernov, N., Markarian, R.: Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28, 271–314 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chernov, N., Markarian, R.: Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28, 315–342 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chernov, N., Markarian, R.: Chaotic Billiards. Mathematical Surveys and Monographs, vol. 127, AMS, Providence, RI (2006)

    Google Scholar 

  12. Chernov, N., van den Bedem, H.: Expanding maps of an interval with holes. Ergod. Th. Dynam. Sys. 22, 637–654 (2002)

    MATH  Google Scholar 

  13. Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlinearity 18, 1527–1553 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chernov, N., Markarian, R., Troubetskoy, S.: Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. Dynam. Sys. 18, 1049–1073 (1998)

    Article  MATH  Google Scholar 

  15. Chernov, N., Markarian, R., Troubetskoy, S.: Invariant measures for Anosov maps with small holes. Ergod. Th. Dynam. Sys. 20, 1007–1044 (2000)

    Article  MATH  Google Scholar 

  16. Collet, P., Martínez, S., Schmitt, B.: The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7, 1437–1443 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Collet, P., Martínez, S., Schmitt, B.: Quasi-stationary distribution and Gibbs measure of expanding systems. In: Tirapegui, E., Zeller, W. (eds.) Instabilities and Nonequilibrium Structures V, pp. 205–19. Kluwer, Dordrecht (1996)

    Chapter  Google Scholar 

  18. Demers, M.F.: Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval. Israel J. Math. 146, 189–221 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Demers, M.F.: Markov Extensions and Conditionally Invariant Measures for Certain Logistic Maps with Small Holes. Ergod. Th. Dynam. Sys. 25(4), 1139–1171 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Demers, M.F., Wright, P.: Behavior of the escape rate function in hyperbolic dynamical systems. Nonlinearity 25, 2133–2150 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Demers, M.F., Young, L.-S.: Escape rates and conditionally invariant measures. Nonlinearity 19, 377–397 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Demers, M.F., Zhang, H.-K.: Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dynam. 5(4), 665–709 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Demers, M.F., Zhang, H.-K.: Spectral analysis for hyperbolic systems with singularities. Nonlinearity 27, 379–433 (2014)

    Article  MathSciNet  Google Scholar 

  24. Demers, M.F., Wright, P., Young, L.-S.: Escape rates and physically relevant measures for billiards with small holes. Commun. Math. Phys. 294(2), 353–388 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Demers, M.F., Wright, P., Young, L.-S.: Entropy, Lyapunov exponents and escape rates in open systems. Ergod. Th. Dynam. Sys. 32(4), 1270–1301 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hennion, H., Hevré, L.: Limit theorems for markov chains and stochastic properties of dynamical systems by quasi-Compactness. Lecture Notes in Mathematics, vol. 1766, Springer, Berlin (2001)

    Google Scholar 

  27. Homburg, A., Young, T.: Intermittency in families of unimodal maps. Ergod. Th. Dynam. Sys. 22(1), 203–225 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Katok, A., Strelcyn, J.-M.: with the collaboration of F. Ledrappier and F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Lecture Notes in Mathematics, vol. 1222, 283 pp. Springer, Berlin (1986)

    Google Scholar 

  29. Keller, G., Liverani, G.: Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28(4), 141–152 (1999)

    MATH  MathSciNet  Google Scholar 

  30. Liverani, C., Maume-Deschamps, V.: Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Annales de l’Institut Henri Poincaré Probab. Stat. 39, 385–412 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lopes, A., Markarian, R.: Open Billiards: Cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56, 651–680 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Pianigiani, G., Yorke, J.: Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252, 351–366 (1979)

    MATH  MathSciNet  Google Scholar 

  33. Richardson, Jr., P.A.: Natural measures on the unstable and invariant manifolds of open billiard dynamical systems. Doctoral Dissertation, Department of Mathematics, University of North Texas, (1999)

    Google Scholar 

  34. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This research is partially supported by NSF grant DMS-1101572.

Author information

Authors and Affiliations

  1. Fairfield University, 1073 N. Benson Road, Fairfield, CT, 06824, USA

    Mark F. Demers

Authors
  1. Mark F. Demers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark F. Demers .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Leicestershire, Leicestershire, United Kingdom

    Wael Bahsoun

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    Christopher Bose

  3. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Gary Froyland

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this paper

Cite this paper

Demers, M.F. (2014). Dispersing Billiards with Small Holes. In: Bahsoun, W., Bose, C., Froyland, G. (eds) Ergodic Theory, Open Dynamics, and Coherent Structures. Springer Proceedings in Mathematics & Statistics, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0419-8_8

Download citation

Publish with us

Policies and ethics

Search

Navigation