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Entropy, Lyapunov exponents and escape rates in open systems

Published online by Cambridge University Press:  06 September 2011

MARK F. DEMERS ,
PAUL WRIGHT  and
LAI-SANG YOUNG
MARK F. DEMERS
Affiliation:
Department of Mathematics and Computer Science, Fairfield University, Fairfield, USA (email: [email protected])
PAUL WRIGHT
Affiliation:
Department of Mathematics, University of Maryland, College Park, USA (email: [email protected])
LAI-SANG YOUNG
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, USA (email: [email protected])
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Abstract

We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions, including Lebesgue and Sinai–Reulle–Bowen (SRB) measures. Lower bounds do not hold in such a generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1270 - 1301
Copyright
Copyright © Cambridge University Press 2011

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References

[A]Abramov, L. M.. The entropy of a derived automorphism. Dokl. Akad. Nauk. SSSR 128 (1959), 647–650; Amer. Math. Soc. Transl. 49(2) (1966), 162–166.Google Scholar
[B]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[BK]Brin, M. and Katok, A.. On local entropy. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 30–38.CrossRefGoogle Scholar
[BDM]Bruin, H., Demers, M. and Melbourne, I.. Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Th. & Dynam. Sys. 30 (2010), 687–728.CrossRefGoogle Scholar
[Bu]Buzzi, J.. Markov extensions for multidimensional dynamical systems. Israel J. Math. 112 (1999), 357–380.CrossRefGoogle Scholar
[CM1]Chernov, N. and Markarian, R.. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28 (1997), 271–314.CrossRefGoogle Scholar
[CM2]Chernov, N. and Markarian, R.. Anosov maps with rectangular holes. Non-ergodic cases. Bol. Soc. Bras. Mat. 28 (1997), 315–342.CrossRefGoogle Scholar
[CM3]Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127). American Mathematical Society, Providence, RI, 2006.Google Scholar
[CMT]Chernov, N., Markarian, R. and Troubetzkoy, S.. Invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 1049–1073.CrossRefGoogle Scholar
[CMS]Collet, P., Martínez, 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
[D]Demers, M. F.. Functional norms for Young towers. Ergod. Th. & Dynam. Sys. 30(5) (2010), 1371–1398.CrossRefGoogle Scholar
[DWY]Demers, M. F., Wright, P. and Young, L.-S.. Escape rates and physically relevant measures for billiards with small holes. Comm. Math. Phys. 294 (2010), 353–388.CrossRefGoogle Scholar
[DY]Demers, M. F. and Young, L.-S.. Escape rates and conditionally invariant measures. Nonlinearity 19 (2006), 377–397.Google Scholar
[KS]Katok, A. and Strelcyn, J.-M.. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, Berlin, 1986, p. 283; with the collaboration of F. Ledrappier and F. Przytycki.CrossRefGoogle Scholar
[LY]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Part I: characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122 (1985), 509–539.CrossRefGoogle Scholar
[M]Mañé, R.. A proof of Pesin’s formula. Ergod. Th. & Dynam. Sys. 1(1) (1981), 95–102.CrossRefGoogle Scholar
[P]Pesin, Ya. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32(4) (1977), 55–114.CrossRefGoogle Scholar
[Pe]Petersen, K. E.. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge University Press, Cambridge, 1989, p. 329.Google Scholar
[Ro]Rohklin, V.. Exact endomorphisms of Lebesgue space. Amer. Math. Soc. Transl. 2 39 (1964), 1–36.Google Scholar
[R]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9(1) (1978), 83–87.CrossRefGoogle Scholar
[S]Sarig, O.. Thermodynamic formalism of countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
[W]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982, p. 250.CrossRefGoogle Scholar
[Y1]Young, L.-S.. Some large deviation results for dynamical systems. Trans. Amer. Math. Soc. 318(2) (1990), 525–543.Google Scholar
[Y2]Young, L.-S.. Ergodic theory of differentiable dynamical systems. Real and Complex Dynamics (NATO ASI Series, C464). Eds. Branner, B. and Hjorth, P.. Kluwer Academic Publishers, Dordrecht, 1995, pp. 293–336.CrossRefGoogle Scholar
[Y3]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.CrossRefGoogle Scholar
[Y4]Young, L.-S.. What are SRB measures and which dynamical systems have them? J. Stat. Phys. 108 (2002), 733–754.CrossRefGoogle Scholar