Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 11
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Kadyrov, Shirali 2015. Exceptional sets in homogeneous spaces and Hausdorff dimension. Dynamical Systems, Vol. 30, Issue. 2, p. 149.

    Froyland, Gary Pollett, Philip K. and Stuart, Robyn M. 2014. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, Vol. 1, Issue. 1, p. 135.

    Mohapatra, Anushaya and Ott, William 2014. Memory loss for nonequilibrium open dynamical systems. Discrete and Continuous Dynamical Systems, Vol. 34, Issue. 9, p. 3747.

    Demers, Mark F. 2013. Escape rates and physical measures for the infinite horizon Lorentz gas with holes. Dynamical Systems, Vol. 28, Issue. 3, p. 393.

    Bahsoun, Wael and Vaienti, Sandro 2012. Metastability of certain intermittent maps. Nonlinearity, Vol. 25, Issue. 1, p. 107.

    Demers, Mark F and Wright, Paul 2012. Behaviour of the escape rate function in hyperbolic dynamical systems. Nonlinearity, Vol. 25, Issue. 7, p. 2133.

    DEMERS, MARK F. WRIGHT, PAUL and YOUNG, LAI-SANG 2012. Entropy, Lyapunov exponents and escape rates in open systems. Ergodic Theory and Dynamical Systems, Vol. 32, Issue. 04, p. 1270.

    Bardet, Jean-Baptiste and Fernandez, Bastien 2011. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete and Continuous Dynamical Systems, Vol. 31, Issue. 3, p. 669.

    BUNDFUSS, STEFAN KRÜGER, TYLL and TROUBETZKOY, SERGE 2011. Topological and symbolic dynamics for hyperbolic systems with holes. Ergodic Theory and Dynamical Systems, Vol. 31, Issue. 05, p. 1305.

    Bunimovich, Leonid A. and Yurchenko, Alex 2011. Where to place a hole to achieve a maximal escape rate. Israel Journal of Mathematics, Vol. 182, Issue. 1, p. 229.

    Froyland, Gary Murray, Rua and Stancevic, Ognjen 2011. Spectral degeneracy and escape dynamics for intermittent maps with a hole. Nonlinearity, Vol. 24, Issue. 9, p. 2435.


Existence and convergence properties of physical measures for certain dynamical systems with holes

  • DOI:
  • Published online: 24 November 2009

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet–Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure μ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of μ under the renormalized dynamics of the system. In addition, we construct an invariant measure ν, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that ν satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet–Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the a.c.c.i.m. and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2]V. Baladi . Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.

[3]V. Baladi and G. Keller . Zeta functions and transfer operators for piecewise monotonic transformations. Comm. Math. Phys. 127 (1990), 459477.

[5]H. Bruin and G. Keller . Equilibrium states for unimodal maps. Ergod. Th. & Dynam. Sys. 18 (1998), 765789.

[8]J. Buzzi . Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357380.

[10]N. N. Cencova . Statistical properties of smooth Smale horseshoes. Mathematical Problems of Statistical Mechanics and Dynamics. Ed. R. L. Dobrushin . Reidel, Dordrecht, 1986, pp. 199256.

[12]N. Chernov and R. Markarian . Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28 (1997), 315342.

[13]N. Chernov , R. Markarian and S. Troubetzkoy . Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 10491073.

[14]N. Chernov , R. Markarian and S. Troubetzkoy . Invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 20 (2000), 10071044.

[15]P. Collet , S. Martinez and B. Schmitt . The Yorke–Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7 (1994), 14371443.

[16]P. Collet , S. Martinez and B. Schmitt . Quasi-stationary distribution and Gibbs measure of expanding systems. Instabilities and Nonequilibrium Structures V. Eds. E. Tirapegui and W. Zeller . Kluwer, Dordrecht, 1996, pp. 205219.

[18]M. Demers . Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. Math. 146 (2005), 189221.

[20]M. Demers and C. Liverani . Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9) (2008), 47774814.

[21]M. Demers and L.-S. Young . Escape rates and conditionally invariant measures. Nonlinearity 19 (2006), 377397.

[22]K. Díaz-Ordaz , M. Holland and S. Luzzatto . Statistical properties of one-dimensional maps with critical points and singularities. Stoch. Dyn. 6 (2006), 423458.

[24]G. Keller . Markov extensions, zeta-functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Amer. Math. Soc. 314 (1989), 433497.

[25]C. Liverani . Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78 (1995), 11111129.

[26]C. Liverani and V. Maume-Deschamps . Lasota–Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), 385412.

[27]A. Lopes and R. Markarian . Open billiards: Cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56 (1996), 651680.

[28]W. de Melo and S. van Strien . One Dimensional Dynamics (Ergebnisse Series, 25). Springer, Berlin, 1993.

[29]G. Pianigiani and J. Yorke . Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351366.

[31]O. Sarig . Thermodynamic formalism of countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 15651593.

[32]L.-S. Young . Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585650.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *