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

Published online by Cambridge University Press:  24 November 2009

HENK BRUIN ,
MARK DEMERS  and
IAN MELBOURNE
HENK BRUIN
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: h.bruin@surrey.ac.uk, ism@math.uh.edu)
MARK DEMERS
Affiliation:
Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT 06824, USA (email: mdemers@fairfield.edu)
IAN MELBOURNE
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: h.bruin@surrey.ac.uk, ism@math.uh.edu)
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Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 687 - 728
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Abramov, L. M.. The entropy of a derived automorphism. Dokl. Akad. Nauk. SSSR 128 (1959), 647–650. Amer. Math. Soc. Transl. Ser. 2 49(2) (1966), 162–166.Google Scholar
[2]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[3]Baladi, V. and Keller, G.. Zeta functions and transfer operators for piecewise monotonic transformations. Comm. Math. Phys. 127 (1990), 459–477.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[5]Bruin, H. and Keller, G.. Equilibrium states for unimodal maps. Ergod. Th. & Dynam. Sys. 18 (1998), 765–789.CrossRefGoogle Scholar
[6]Bruin, H., Luzzatto, S. and van Strien, S.. Decay of correlations in one-dimensional dynamics. Ann. Sci. École Norm. Sup. (4) 36 (2003), 621–646.CrossRefGoogle Scholar
[7]Bruin, H. and Todd, M.. Equilibrium states for interval maps: the potential −tlog ∣Df∣. Ann Sci. École Norm Sup. to appear.Google Scholar
[8]Buzzi, J.. Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357–380.CrossRefGoogle Scholar
[9]Cencova, N. N.. A natural invariant measure on Smale’s horseshoe. Soviet Math. Dokl. 23 (1981), 87–91.Google Scholar
[10]Cencova, N. N.. Statistical properties of smooth Smale horseshoes. Mathematical Problems of Statistical Mechanics and Dynamics. Ed. Dobrushin, R. L.. Reidel, Dordrecht, 1986, pp. 199–256.CrossRefGoogle Scholar
[11]Chernov, N. and Markarian, R.. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28 (1997), 271–314.CrossRefGoogle Scholar
[12]Chernov, N. and Markarian, R.. Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28 (1997), 315–342.CrossRefGoogle Scholar
[13]Chernov, N., Markarian, R. and Troubetzkoy, S.. Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 1049–1073.CrossRefGoogle Scholar
[14]Chernov, N., Markarian, R. and Troubetzkoy, S.. Invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 20 (2000), 1007–1044.CrossRefGoogle Scholar
[15]Collet, P., Martinez, S. and Schmitt, B.. The Yorke–Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7 (1994), 1437–1443.CrossRefGoogle Scholar
[16]Collet, P., Martinez, S. and Schmitt, B.. Quasi-stationary distribution and Gibbs measure of expanding systems. Instabilities and Nonequilibrium Structures V. Eds. Tirapegui, E. and Zeller, W.. Kluwer, Dordrecht, 1996, pp. 205–219.CrossRefGoogle Scholar
[17]Chernov, N. and van den Bedem, H.. Expanding maps of an interval with holes. Ergod. Th. & Dynam. Sys. 22 (2002), 637–654.Google Scholar
[18]Demers, M.. Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. Math. 146 (2005), 189–221.CrossRefGoogle Scholar
[19]Demers, M.. Markov extensions and conditionally invariant measures for certain logistic maps with small holes. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1139–1171.CrossRefGoogle Scholar
[20]Demers, M. and Liverani, C.. Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9) (2008), 4777–4814.CrossRefGoogle Scholar
[21]Demers, M. and Young, L.-S.. Escape rates and conditionally invariant measures. Nonlinearity 19 (2006), 377–397.CrossRefGoogle Scholar
[22]Díaz-Ordaz, K., Holland, M. and Luzzatto, S.. Statistical properties of one-dimensional maps with critical points and singularities. Stoch. Dyn. 6 (2006), 423–458.CrossRefGoogle Scholar
[23]Homburg, A. and Young, T.. Intermittency in families of unimodal maps. Ergod. Th. & Dynam. Sys. 22(1) (2002), 203–225.CrossRefGoogle Scholar
[24]Keller, G.. Markov extensions, zeta-functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Amer. Math. Soc. 314 (1989), 433–497.CrossRefGoogle Scholar
[25]Liverani, C.. Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78 (1995), 1111–1129.CrossRefGoogle Scholar
[26]Liverani, C. and Maume-Deschamps, V.. 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), 385–412.CrossRefGoogle Scholar
[27]Lopes, A. and Markarian, R.. Open billiards: Cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56 (1996), 651–680.CrossRefGoogle Scholar
[28]de Melo, W. and van Strien, S.. One Dimensional Dynamics (Ergebnisse Series, 25). Springer, Berlin, 1993.CrossRefGoogle Scholar
[29]Pianigiani, G. and Yorke, J.. Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351–366.Google Scholar
[30]Richardson, P. A. Jr. 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
[31]Sarig, O.. Thermodynamic formalism of countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
[32]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585–650.CrossRefGoogle Scholar