Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 555-593.

Nous prouvons l'existence d'un nombre fini d'états d' équilibre ergodiques pour une classe assez grande d'homéomorphismes locaux non-uniformément dilatants sur des espaces métriques compacts et pour les potentiels de Hölder continus à oscillation pas trop grande. Aucune structure de Markov n'est supposée. Si la transformation est topologiquement mélangeante alors il existe un unique état d' équilibre, il est une mesure exacte et vérifie une propriété de Gibbs non-uniforme. Avec quelques hypothèses supplémentaires, nous prouvons aussi que les états d' équilibre varient de façon continue avec la dynamique et le potentiel (stabilité statistique) et sont également stables sous des perturbations stochastiques de la transformation.

We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact metric spaces and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation.

@article{AIHPC_2010__27_2_555_0,
     author = {Varandas, Paulo and Viana, Marcelo},
     title = {Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {555--593},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.10.002},
     mrnumber = {2595192},
     zbl = {1193.37009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.10.002/}
}
TY  - JOUR
AU  - Varandas, Paulo
AU  - Viana, Marcelo
TI  - Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 555
EP  - 593
VL  - 27
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.10.002/
DO  - 10.1016/j.anihpc.2009.10.002
LA  - en
ID  - AIHPC_2010__27_2_555_0
ER  - 
%0 Journal Article
%A Varandas, Paulo
%A Viana, Marcelo
%T Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 555-593
%V 27
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.10.002/
%R 10.1016/j.anihpc.2009.10.002
%G en
%F AIHPC_2010__27_2_555_0
Varandas, Paulo; Viana, Marcelo. Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 555-593. doi : 10.1016/j.anihpc.2009.10.002. http://www.numdam.org/articles/10.1016/j.anihpc.2009.10.002/

[1] J.F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (2000), 331-332 | EuDML | MR

[2] J.F. Alves, V. Araújo, Random perturbations of nonuniformly expanding maps, Astérisque 286 (2003), 25-62 | Numdam | MR | Zbl

[3] J.F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351-398 | MR | Zbl

[4] V. Araújo, Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures, Discrete Contin. Dyn. Syst. 17 (2007), 371-386 | MR | Zbl

[5] A. Arbieto, C. Matheus, Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials, preprint http://www.preprint.impa.br, 2006

[6] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. vol. 470, Springer-Verlag (1975) | MR | Zbl

[7] R. Bowen, D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), 181-202 | EuDML | MR | Zbl

[8] H. Bruin, G. Keller, Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems 18 (1998), 765-789 | MR | Zbl

[9] H. Bruin, M. Todd, Equilibrium states for potentials with sup ϕ- inf ϕ<h top (f), Commun. Math. Phys. 283 (2008), 579-611 | Zbl

[10] H. Bruin, M. Todd, Equilibrium states for interval maps: The potential -t log |Df|, Ann. Sci. École Norm. Sup., in press | EuDML | MR

[11] J. Buzzi, Markov extensions for multi-dimensional dynamical systems, Israel J. Math. 112 (1999), 357-380 | MR | Zbl

[12] J. Buzzi, On entropy-expanding maps, preprint, 2000

[13] J. Buzzi, No or infinitely many a.c.i.p. for piecewise expanding c r maps in higher dimensions, Commun. Math. Phys. 222 (2001), 495-501 | MR | Zbl

[14] J. Buzzi, Subshifts of quasi-finite type, Invent. Math. 159 no. 2 (2005), 369-406 | MR | Zbl

[15] J. Buzzi, V. Maume-Deschamps, Decay of correlations for piecewise invertible maps in higher dimensions, Israel J. Math. 131 (2002), 203-220 | MR | Zbl

[16] J. Buzzi, F. Paccaut, B. Schmitt, Conformal measures for multidimensional piecewise invertible maps, Ergodic Theory Dynam. Systems 21 (2001), 1035-1049 | MR | Zbl

[17] J. Buzzi, S. Ruette, Large entropy implies existence of a maximal entropy measure for interval maps, Discrete Contin. Dyn. Syst. 14 (2006), 673-688 | MR | Zbl

[18] J. Buzzi, O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems 23 (2003), 1383-1400 | MR | Zbl

[19] M. De Guzman, Differentiation of Integrals in n , Lecture Notes in Math. vol. 481, Springer-Verlag (1975) | MR

[20] M. Denker, M. Urbański, Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), 561-579 | EuDML | MR | Zbl

[21] M. Denker, M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134 | MR | Zbl

[22] M. Denker, M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. 43 (1991), 107-118 | MR | Zbl

[23] M. Denker, M. Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), 563-587 | MR | Zbl

[24] M. Denker, M. Urbański, The dichotomy of Hausdorff measures and equilibrium states for parabolic rational maps, Ergodic Theory and Related Topics, III, Güstrow, 1990, Lecture Notes in Math. vol. 1514, Springer (1992), 90-113 | MR | Zbl

[25] M. Denker, G. Keller, M. Urbański, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math. 97 (1990), 27-36 | EuDML | MR | Zbl

[26] M. Denker, Z. Nitecki, M. Urbański, Conformal measures and S-unimodal maps, Dynamical Systems and Applications, World Sci. Ser. Appl. Anal. vol. 4, World Sci. Publ. (1995), 169-212 | MR | Zbl

[27] M. Denker, F. Przytycki, M. Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), 255-266 | MR | Zbl

[28] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. vol. 153, Springer-Verlag (1969) | MR | Zbl

[29] V. Horita, M. Viana, Hausdorff dimension for non-hyperbolic repellers. II. DA diffeomorphisms, Discrete Contin. Dyn. Syst. 13 (2005), 1125-1152 | MR | Zbl

[30] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems 1 (1981), 77-93 | MR | Zbl

[31] F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Publ. Math. Inst. Hautes Etudes Sci. 59 (1984), 163-188 | EuDML | Numdam | MR | Zbl

[32] R. Leplaideur, I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes, Nonlinearity 19 (2006), 2667-2694 | MR | Zbl

[33] R. Leplaideur, I. Rios, On t-conformal measures and Hausdorff dimension for a family non-uniformly hyperbolic horseshoes, Ergodic Theory Dynam. Systems, in press | MR

[34] P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J. 150 (1998), 197-209 | MR | Zbl

[35] R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems 1 (1981), 95-101 | MR | Zbl

[36] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag (1987) | MR

[37] K. Oliveira, Equilibrium states for certain non-uniformly hyperbolic systems, Ergodic Theory Dynam. Systems 23 (2003), 1891-1906 | MR

[38] K. Oliveira, M. Viana, Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms, Discrete Contin. Dyn. Syst. 15 (2006), 225-236 | MR | Zbl

[39] K. Oliveira, M. Viana, Thermodynamical formalism for an open classes of potentials and non-uniformly hyperbolic maps, Ergodic Theory Dynam. Systems 28 (2008) | MR | Zbl

[40] W. Parry, Entropy and Generators in Ergodic Theory, W.A. Benjamin (1969) | MR | Zbl

[41] Ya. Pesin, Dimension Theory in Dynamical Systems, University of Chicago Press (1997) | MR

[42] Ya. Pesin, S. Senti, Thermodynamical formalism associated with inducing schemes for one-dimensional maps, Mosc. Math. J. 5 no. 743 (2005), 669-678 | MR | Zbl

[43] Ya. Pesin, S. Senti, K. Zhang, Lifting measures to inducing schemes, Ergodic Theory Dynam. Systems 28 no. 2 (2008), 553-574 | MR | Zbl

[44] V. Pinheiro, Expanding measures, preprint, 2008 | Numdam | MR

[45] M. Qian, S. Zhu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. Amer. Math. Soc. 354 (2002), 1453-1471 | MR | Zbl

[46] V.A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499-530 | MR

[47] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976), 619-654 | MR | Zbl

[48] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), 1565-1593 | MR | Zbl

[49] O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), 555-577 | MR | Zbl

[50] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), 1751-1758 | MR | Zbl

[51] Ya. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys 27 (1972), 21-69 | MR | Zbl

[52] M. Urbański, Hausdorff measures versus equilibrium states of conformal infinite iterated function systems, International Conference on Dimension and Dynamics Miskolc, 1998 Period. Math. Hungar. 37 (1998), 153-205 | MR | Zbl

[53] P. Varandas, Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps, J. Stat. Phys. 133 (2008), 813-839 | MR | Zbl

[54] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag (1982) | MR | Zbl

[55] Q. Wang, L.-S. Young, Strange attractors with one direction of instability, Comm. Math. Phys. 218 (2001), 1-97 | Zbl

[56] M. Yuri, Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory Dynam. Systems 19 (1999), 1365-1378 | MR | Zbl

[57] M. Yuri, Weak Gibbs measures for certain non-hyperbolic systems, Ergodic Theory Dynam. Systems 20 (2000), 1495-1518 | MR | Zbl

[58] M. Yuri, Thermodynamical formalism for countable to one Markov systems, Trans. Amer. Math. Soc. 335 (2003), 2949-2971 | MR | Zbl

Cité par Sources :