Skip to main content
Springer Nature Link
Log in

Markov extensions for dynamical systems with holes: An application to expanding maps of the interval

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. forC 1+α expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [BC] H. van den Bedem and N. Chernov,Expanding maps of an interval with holes, Ergodic Theory and Dynamical Systems22 (2002), 637–654.

    Article  MATH  MathSciNet  Google Scholar 

  • [BK] V. Baladi and G. Keller,Zeta functions and transfer operators for piecewise monotonic transformations, Communications in Mathematical Physics127, (1990), 459–477.

    Article  MATH  MathSciNet  Google Scholar 

  • [BY] M. Benedicks and L.-S. Young,Markov extensions and decay of correlations for certain Hénon maps, Asterisque261 (2000), xi, 13–56.

    MathSciNet  Google Scholar 

  • [C1] N. N. Čencova,A natural invariant measure on Smale's horseshoe, Soviet Mathematics Doklady23 (1981), 87–91.

    Google Scholar 

  • [C2] N. N. Čencova,Statistical properties of smooth Smale horseshoes, inMathematical Problems of Statistical Mechanics and Dynamics (R. L. Dobrushin, ed.), Reidel, Dordrecht, 1986, pp. 199–256.

    Google Scholar 

  • [Ch1] N. Chernov,Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete and Continuous Dynamical Systems5 (1999), 425–448.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ch2] N. Chernov,Decay of correlations and dispersing billiards, Journal of Statistical Physics94 (1999), 513–556.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ch3] N. Chernov,Sinai billiards under small external forces, Annales de l'Institut Henri Poincaré2 (2001), 197–236.

    Article  MATH  MathSciNet  Google Scholar 

  • [CM1] N. Chernov and R. Markarian,Argodic properties of Anosov maps with rectangular holes, Boletim da Sociedade Brasileira de Matemática28 (1997), 271–314.

    Article  MATH  MathSciNet  Google Scholar 

  • [CM2] N. Chernov and R. Markarian,Anosov maps with rectangular holes. Nonergodic cases, Boletim da Sociedade Brasileira de Matemática28 (1997), 315–342.

    Article  MATH  MathSciNet  Google Scholar 

  • [CMS1] P. Collet, S. Martínez and B. Schmitt,The Yorke—Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems Nonlinearity7 (1994), 1437–1443.

    Article  MATH  MathSciNet  Google Scholar 

  • [CMS2] P. Collet, S. Martínez and B. Schmitt,Quasi-stationary distribution and Gibbs measure of expanding systems, inInstabilities and Nonequilibrium Structures V (E. Tirapegui and W. Zeller, eds.), Kluwer, Dordrecht, 1996, pp. 205–219.

    Google Scholar 

  • [CMT1] N. Chernov, R. Markarian and S. Troubetskoy,Conditionally invariant measures for Anosov maps with small holes, Ergodic Theory and Dynamical Systems18 (1998), 1049–1073.

    Article  MATH  MathSciNet  Google Scholar 

  • [CMT2] N. Chernov, R. Markarian and S. Troubetskoy,Invariant measures for Anosov maps with small holes, Ergodic Theory and Dynamical Systems20 (2000), 1007–1044.

    Article  MATH  MathSciNet  Google Scholar 

  • [D] M. Demers,Markov extensions and conditionally invariant measures for certain logistic maps with small holes, Ergodic Theory and Dynamical Systems, to appear.

  • [KL] G. Keller and C. Liverani,Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa Cl. Sci. (4)28 (1999), 141–152.

    MATH  MathSciNet  Google Scholar 

  • [LiM] C. Liverani and V. Maume-Deschamps,Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set, Annales de l'Institut Henri Poincaré. Probabilités et Statistiques39 (2003), 385–412.

    Article  MATH  MathSciNet  Google Scholar 

  • [LM] A. Lopes and R. Markarian,Open Billiards: Cantor sets, invariant and conditionally invariant probabilities, SIAM Journal of Applied Mathematics56 (1996), 651–680.

    Article  MATH  MathSciNet  Google Scholar 

  • [PY] G. Pianigiani and J. Yorke,Expanding maps on sets which are almost invariant: decay and chaos, Transactions of the American Mathematical Society252 (1979), 351–366.

    Article  MATH  MathSciNet  Google Scholar 

  • [R] P. A. Richardson, Jr.,Natural Measures on the Unstable and Invariant Manifolds of Open Billiard Dynamical Systems, Doctoral Dissertation, Department of Mathematics, University of North Texas, 1999.

  • [Y1] L.-S. Young,Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics147 (1998), 585–650.

    Article  MATH  MathSciNet  Google Scholar 

  • [Y2] L.-S. Young,Recurrence times and rates of mixing, Israel Journal of Mathematics110 (1999), 153–188.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Georgia Institute of Technology, 30332, Atlanta, GA, 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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demers, M.F. Markov extensions for dynamical systems with holes: An application to expanding maps of the interval. Isr. J. Math. 146, 189–221 (2005). https://doi.org/10.1007/BF02773533

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773533

Keywords

Search

Navigation