Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

Castro, A.; Varandas, P.

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 225-249.

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle–Perron–Frobenius operator acting on the space of Hölder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

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

[2] V. Baladi, L.S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys. 156 (1993), 355-385 | MR | Zbl

[3] V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific Publishing Co. Inc. (2000) | MR | Zbl

[4] V. Baladi, S.S. Gouezel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009), 1453-1481 | EuDML | Numdam | MR | Zbl

[5] V. Baladi, S.S. Gouezel, Banach spaces for piecewise cone hyperbolic maps, J. Mod. Dyn. 4 (2010), 91-137 | MR | Zbl

[6] V. Baladi, D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity 21 (2008), 677-711 | MR | Zbl

[7] V. Baladi, D. Smania, Analyticity of the SRB measure for holomorphic families of quadratic-like Collet–Eckmann maps, Proc. Amer. Math. Soc. 137 (2009), 1431-1437 | MR | Zbl

[8] V. Baladi, M. Tsujii, Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier 57 (2007), 27-54 | Numdam | MR

[9] M. Blank, G. Keller, C. Liverani, Ruelle–Perron–Frobenius spectrum for Anosov maps, Nonlinearity 15 (2001), 1905-1973 | MR | Zbl

[10] T. Bomfim, A. Castro, P. Varandas, Linear response formula for equilibrium states in non-uniformly expanding dynamics, arXiv:1205.5361 (2012) | MR | Zbl

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

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

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

[14] H. Bruin, M. Todd, Equilibrium states for interval maps: potentials with $\sup \varphi -\inf \varphi <h_{\mathrm{top}}\left(f\right)$, Comm. Math. Phys. 283 (2008), 579-611 | 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, 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

[17] J. Buzzi, T. Fisher, Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems, arXiv:0903.3692

[18] A. Castro, Backward inducing and exponential decay of correlations for partially hyperbolic attractors, Israel J. Math. 130 (2002), 29-75 | MR | Zbl

[19] A. Castro, Fast mixing for attractors with mostly contracting central direction, Ergodic Theory Dynam. Systems 24 (2004), 17-44 | MR | Zbl

[20] M. Demers, C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc. 360 (2008), 4777-4814 | MR | Zbl

[21] 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

[22] D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004), 389-449 | MR | Zbl

[23] S. Gouezel, C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems 26 (2006), 189-217 | MR | Zbl

[24] H. Hennion, L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Math. vol. 1766 (2001) | MR | Zbl

[25] T. Hunt, R. Mackay, Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor, Nonlinearity 16 (2003), 1499-1510 | MR | Zbl

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

[27] C. Liverani, Decay of correlations, Ann. of Math. 142 (1995), 239-301 | MR | Zbl

[28] C. Liverani, B. Saussol, S. Vaienti, Conformal measure and decay of correlation for covering weighted systems, Ergodic Theory Dynam. Systems 18 no. 6 (1998), 1399-1420 | MR | Zbl

[29] 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

[30] Ya. Pesin, S. Senti, K. Zhang, Lifting measures to inducing schemes, Ergodic Theory Dynam. Systems 28 (2008), 553-574 | MR | Zbl

[31] V. Pinheiro, Expanding measures, Ann. Inst. H. Poincare Anal. Non Lineaire 28 (2011), 889-939 | Numdam | Zbl

[32] V. Pinheiro, P. Varandas, Thermodynamical formalism for expanding measures, preprint, UFBA.

[33] F. Przytycki, J. Rivera-Letelier, Statistical properties of topological Collet–Eckmann maps, Ann. Sci. Ec. Norm. Super. 40 (2007), 135-178 | EuDML | Numdam | MR | Zbl

[34] F. Przytycki, J. Rivera-Letelier, Nice inducing schemes and the thermodynamics of rational maps, Comm. Math. Phys. 301 no. 3 (2011), 661-707 | MR | Zbl

[35] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187 (1997), 227-241 | MR | Zbl

[36] H.H. Rugh, Cones and gauges in complex spaces: spectral gaps and complex Perron–Frobenius theory, Ann. of Math. 171 (2010), 1707-1752 | MR | Zbl

[37] M. Sambarino, C. Vásquez, Bowen measure for derived from Anosov diffeomorphisms, arXiv:0904.1036

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

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

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

[41] P. Varandas, M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010), 555-593 | Numdam | MR | Zbl

[42] M. Viana, Stochastic Dynamics of Deterministic Systems, Colóquio Brasileiro de Matemática, IMPA (1997)

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

[44] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147 no. 3 (1998), 585-650 | MR | Zbl