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Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps

Published online by Cambridge University Press:  01 April 2008

KRERLEY OLIVEIRA and
MARCELO VIANA
KRERLEY OLIVEIRA
Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, 57072-090 Maceio, Brazil (email: krerley@gmail.com)
MARCELO VIANA
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110 - Jardim Botanico, 22460-320 Rio de Janeiro, Brazil (email: viana@impa.br)
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Abstract

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We develop a Ruelle–Perron–Fröbenius transfer operatorapproach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Hölder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, which is piecewise Hölder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008, pp. 501 - 533
Copyright
Copyright © Cambridge University Press 2008

References

[1]Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.CrossRefGoogle Scholar
[2]Alves, J. F.. SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. École Norm. Sup. 33 (2000), 1–32.CrossRefGoogle Scholar
[3]Arbieto, A. and Matheus, C.. Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials. Preprint www.preprint.impa.br, 2006.Google Scholar
[4]Arbieto, A., Matheus, C. and Oliveira, K.. Equilibrium states for random non-uniformly expanding maps. Nonlinearity 17 (2004), 581–593.CrossRefGoogle Scholar
[5]Bruin, H. and Keller, G.. Equilibrium states for S-unimodal maps. Ergod. Th. & Dynam. Sys. 18 (1998), 765–789.CrossRefGoogle Scholar
[6]Buzzi, J. and Maume-Deschamps, V.. Decay of correlations for piecewise invertible maps in higher dimensions. Israel J. Math. 131 (2002), 203–220.CrossRefGoogle Scholar
[7]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[8]Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[9]Buzzi, J. and Ruette, S.. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete Contin. Dyn. Syst. 14 (2006), 673–688.CrossRefGoogle Scholar
[10]Buzzi, J. and Sarig, O.. Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod. Th. & Dynam. Sys. 23 (2003), 1383–1400.CrossRefGoogle Scholar
[11]Buzzi, J.. Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357–380.CrossRefGoogle Scholar
[12]Buzzi, J.. Subshifts of quasi-finite type. Invent. Math. 159(2) (2005), 369–406.CrossRefGoogle Scholar
[13]Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.CrossRefGoogle Scholar
[14]Deimling, K.. Nonlinear Functional Analysis. Springer, Berlin, 1985.CrossRefGoogle Scholar
[15]Denker, M. and Urbański, M.. Ergodic theory of equilibrium states for rational maps. Nonlinearity 4 (1991), 103–134.CrossRefGoogle Scholar
[16]Denker, M. and Urbański, M.. The dichotomy of Hausdorff measures and equilibrium states for parabolic rational maps. Ergodic Theory and Related Topics (Lecture Notes in Mathematics, 1514). Springer, Berlin, 1992, pp. 90–113.CrossRefGoogle Scholar
[17]Hofbauer, F. and Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.CrossRefGoogle Scholar
[18]Kingman, J.. The ergodic theorem of subadditive stochastic processes. J. Roy. Statist. Soc. 30 (1968), 499–510.Google Scholar
[19]Leplaideur, R. and Rios, I.. Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes. Nonlinearity 19 (2006), 2667–2694.CrossRefGoogle Scholar
[20]Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.CrossRefGoogle Scholar
[21]Oliveira, K.. Equilibrium states for certain non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 23 (2003), 1891–1906.CrossRefGoogle Scholar
[22]Oliveira, K. and Viana, M.. Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms. Discrete Contin. Dyn. Syst. 15 (2006), 225–236.CrossRefGoogle Scholar
[23]Parry, W.. Equilibrium states and wieghted uniform distributions of closed orbits. Dynamical Systems—Marylan 86/87, Vol. 1342. Springer, Berlin, 1988, pp. 617–625.CrossRefGoogle Scholar
[24]Pesin, Ya.. Contemporary views and applications. Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago, IL, 1997.CrossRefGoogle Scholar
[25]Pliss, V.. On a conjecture due to Smale. Differ. Uravn. 8 (1972), 262–268.Google Scholar
[26]Przytycki, F., Rivera-Letelier, J. and Smirnov, S.. Thermodynamics and multifractal analysis of non uniformly hyperbolic rational maps. In preparation.Google Scholar
[27]Pesin, Ya. and Senti, S.. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J. 5 (2005), 669–678; 743–744.CrossRefGoogle Scholar
[28]Ruelle, D.. A measure associated with Axiom A attractors. Amer. J. Math. 98 (1976), 619–654.CrossRefGoogle Scholar
[29]Ruelle, D.. An inequality for the entropy of differentiable maps. Bull. Braz. Math. Soc. 9 (1978), 83–87.CrossRefGoogle Scholar
[30]Ruelle, D.. The thermodynamical formalism for expanding maps. Comm. Math. Phys. 125 (1989), 239–262.CrossRefGoogle Scholar
[31]Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
[32]Sarig, O.. Phase transitions for countable Markov shifts. Comm. Math. Phys. 217 (2001), 555–577.CrossRefGoogle Scholar
[33]Sarig, O.. Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131 (2003), 1751–1758 (electronic).CrossRefGoogle Scholar
[34]Sinai, Ya.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (1972), 21–69.CrossRefGoogle Scholar
[35]Urbański, M.. Hausdorff measures versus equilibrium states of conformal infinite iterated function systems. Period. Math. Hungar. 37 (1998), 153–205. (International Conference on Dimension and Dynamics (Miskolc, 1998).)CrossRefGoogle Scholar
[36]Walters, P.. An Introduction to Ergodic Theory. Springer, Berlin, 1982.CrossRefGoogle Scholar
[37]Wang, Q. and Young, L.-S.. Strange attractors with one direction of instability. Comm. Math. Phys. 218 (2001), 1–97.CrossRefGoogle Scholar
[38]Yuri, M.. Thermodynamic formalism for certain nonhyperbolic maps. Ergod. Th. & Dynam. Sys. 19 (1999), 1365–1378.CrossRefGoogle Scholar
[39]Yuri, M.. Weak Gibbs measures for certain non-hyperbolic systems. Ergod. Th. & Dynam. Sys. 20 (2000), 1495–1518.CrossRefGoogle Scholar
[40]Yuri, M.. Thermodynamical formalism for countable to one Markov systems. Trans. Amer. Math. Soc. 335 (2003), 2949–2971.CrossRefGoogle Scholar