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Some statistical properties of almost Anosov diffeomorphisms. (English) Zbl 1448.37039

Summary: For a kind of almost Anosov diffeomorphisms, we study the relationship among the existence of Sinai-Ruelle-Bowen (SRB) measures, the local differentiability near the indifferent fixed points, and space dimension, where the almost Anosov diffeomorphisms are hyperbolic everywhere except for the indifferent fixed points. As a consequence, there are \(C^2\) almost Anosov diffeomorphisms that admit \(\sigma\)-finite (infinite) SRB measures in spaces with dimensions bigger than one; there exist \(C^2\) almost Anosov diffeomorphisms with finite SRB measures in spaces with dimensions bigger than three. Further, we obtain the lower and upper polynomial bounds for the decay rates of the correlation functions of the Hölder observables for the maps admitting finite SRB measures.

MSC:

37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C05 Dynamical systems involving smooth mappings and diffeomorphisms

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[1] Aaronson, J.; Denker, M.; Urbanski, M., Ergodic theory for Markov fibred systems and parabolic rational maps, Trans Amer Math Soc, 337, 495-548 (1993) · Zbl 0789.28010
[2] Alves, J.; Bonatti, C.; Viana, M., SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent Math, 140, 351-398 (2000) · Zbl 0962.37012
[3] Alves, J.; Dias, C.; Luzzatto, S.; Pinheiro, V., SRB measures for partially hyperbolic systems whose central direction is weakly expanding, J. Eur. Math. Society, 19, 2911-2946 (2017) · Zbl 1378.37046
[4] Alves, J.; Leplaideur, R., SRB measures for almost axiom a diffeomorphisms, Ergod Theory Dyn Sys, 36, 2015-2043 (2016) · Zbl 1370.37046
[5] Araújo, V.; Melbourne, I.; Varandas, P., Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps, Commun Math Phys, 340, 901-938 (2015) · Zbl 1356.37002
[6] Araújo, V.; Pacifico, M.; Pujals, E.; Viana, M., Singular-hyperbolic attractors are chaotic, Trans Am Math Soc, 361, 2431-2485 (2009) · Zbl 1214.37010
[7] Baladi, V., Positive transfer operators and decay of correlations, Adv Ser Nonlinear Dyn, 16 (2000) · Zbl 1012.37015
[8] Baladi, V.; Liverani, C., Exponential decay of correlations for piecewise cone hyperbolic contact flows, Commun Math Phys, 314, 689-773 (2012) · Zbl 1386.37030
[9] Bálint, P.; Melbourne, I., Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows, J Stat Phys, 133, 435-447 (2008) · Zbl 1161.82016
[10] Barreira, L.; Pesin, Y., Lectures on Lyapunov exponents and smooth ergodic theory, University Lecture Series, 23 (2002)
[11] Benedicks, M.; Young, L., Sinai-Bowen-Ruelle measure for certain Hénon maps, Invent Math, 112, 541-576 (1993) · Zbl 0796.58025
[12] Benedicks, M.; Young, L., Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261, 13-56 (2000) · Zbl 1044.37013
[13] Bessa, M.; Rocha, J.; Varandas, P., Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles, Dyn Syst, 33, 691-707 (2018) · Zbl 1402.37038
[14] Bochi, J.; Bonatti, C., Perturbation of the Lyapunov spectra of periodic orbits, Proc Lond Math Soc, 105, 1-48 (2012) · Zbl 1268.37027
[15] Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorhisms, Lecture notes in mathematics, 470 (1975), Springer: Springer New York · Zbl 0308.28010
[16] Bowen, R., Markov partitions are not smooth, Proc Am Math Soc, 71, 130-132 (1978) · Zbl 0417.58011
[17] Boyarsky, A.; Góra, P., Laws of chaos: invariant measures and dynamical systems in one dimension (1997), Birkhäuser: Birkhäuser Boston · Zbl 0893.28013
[18] Cawley, E., Smooth Markov partitions and toral automorphisms, Ergod Theory Dyn Syst, 11, 633-651 (1991) · Zbl 0754.58028
[19] Chernov, N.; Markarian, R., Chaotic billiards, Math Surveys Monogr Am Math Soc, 127 (2006) · Zbl 1101.37001
[20] Furstenberg, H., Recurrence in Ergodic theory and Combinaorial number theory (1981), Princeton university press: Princeton university press Princeton, New Jersey · Zbl 0459.28023
[21] Gouëzel, S., Sharp polynomial estimates for the decay of correlations, Israel J Math, 139, 29-65 (2004) · Zbl 1070.37003
[22] Hatomoto, J., Decay of correlations for some partially hyperbolic diffeomorphisms, Hokkaido Math J, 38, 39-65 (2009) · Zbl 1177.37016
[23] Hirsch M, Pugh C. Stable manifolds and hyperbolic sets. Proceedings of the Symposia in Pure Mathematics AMS1970; 14:133-163.; Hirsch M, Pugh C. Stable manifolds and hyperbolic sets. Proceedings of the Symposia in Pure Mathematics AMS1970; 14:133-163. · Zbl 0215.53001
[24] Hirsch, M.; Pugh, C.; Shub, M., Invariant manifolds, Lecture notes in mathematics (1977), Springer: Springer Berlin · Zbl 0355.58009
[25] Hu, H., Conditions for the existence of SBR measures for “almost Anosov” diffeomorphisms, Trans Am Math Soc, 352, 2331-2367 (1999) · Zbl 0949.37007
[26] Hu, H.; Vaienti, S., Absolutely continuous invariant measures for some non-uniformly expanding maps, Ergod Theory Dyn Syst, 29, 1185-1215 (2009) · Zbl 1201.37058
[27] Hu, H.; Young, L., Nonexistence of SBR measures for some diffeomorphisms that are “almost Anosov”, Ergod Theory Dyn Syst, 15, 67-76 (1995) · Zbl 0818.58035
[28] Katok, A.; Strelcyn, J.; Ledrappier, A.; Przytycki, F., Invariant manifolds, entropy and billiards, Smooth Maps with Singularities, Vol. 1222 of Lecture Notes in Mathematics (1986), Springer-Verlag: Springer-Verlag Berlin, Heidelber, New York, London, Paris, Tokyo · Zbl 0658.58001
[29] Ledrappier, F., Propriétés Ergodiques des measure de Sinai, Publ Math IHES, 59, 163-188 (1984) · Zbl 0561.58037
[30] Ledrappier, F.; Strelcyn, J., Estimation from below in Pesin’s entropy formula, Ergod Theory Dyn Sys, 2, 203-219 (1982) · Zbl 0533.58022
[31] Leplaideur, R., Existence of SRB-measures for some topologically hyperbolic diffeomorphisms, Ergod Theory Dyn Sys, 24, 1199-1225 (2004) · Zbl 1062.37012
[32] Liverani, C.; Martens, M., Convergence to equilibrium for intermittent symplectic maps, Commun Math Phys, 260, 527-556 (2005) · Zbl 1094.37034
[33] Mañé, R., Ergodic theory and differentiable dynamics, Translated from the Portuguese by Silvio Levy, Ergebnisse der Mathematik und ihrer Grenzgebiete (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0616.28007
[34] Melbourne, I., Rapid decay of correlations for nonuniformly hyperbolic flows, Trans Am Math Soc, 359, 2421-2441 (2006) · Zbl 1184.37024
[35] Melbourne, I., Decay of correlations for slowly mixing flows, Proc Lond Math Soc, 98, 163-190 (2009) · Zbl 1158.37005
[36] Melbourne, I.; Terhesiu, D., Decay of correlation for nonuniformly expanding systems with general return times, Ergod Theory Dyn Sys, 34, 893-918 (2014) · Zbl 1332.37025
[37] Melbourne, I.; Török, A., Convergence of moments for Axiom A and non-uniformly hyperbolic flows, Ergod Theory Dyn Sys, 32, 1091-1100 (2012) · Zbl 1263.37047
[38] Pesin, Y., Families of invariant manifolds corresponding to non-zero characteristics exponenets, Math USSR-Izv, 10, 1261-1305 (1978) · Zbl 0383.58012
[39] Pianigiani, G., First return map and invariant measures, Israel J Math, 35, 32-48 (1980) · Zbl 0445.28016
[40] Pugachev, V.; Sinitsyn, I., Lectures on functional analysis and applications (1999), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 0931.46001
[41] Robinson, C., Dynamical systems: stability, symbolic dynamics and chaos (1999), CRC Press: CRC Press Florida · Zbl 0914.58021
[42] Rohlin, V., Lectures on the theory of entropy of transformations with invariant measures, Russ Math Surveys, 22, 1-54 (1967)
[43] Saks, S., Theory of the integral (1937), Polish Mathematical Society · Zbl 0017.30004
[44] Sarig, O., Subexponential decay of corrlations, Invent Math, 150, 629-653 (2002) · Zbl 1042.37005
[45] Sarig, O., Introduction to the transfer operator method, Second Brazilian School on Dynamical Systems. Second Brazilian School on Dynamical Systems, Lecture Notes (2012)
[46] Sinai, Y., Gibbs measures in ergodic theory, Russ Math Surveys 27, 4, 21-69 (1972) · Zbl 0255.28016
[47] Snavely, M., Markov partitions for the two-dimensional torus, Proc Am Math Soc, 113, 517-527 (1991) · Zbl 0738.58036
[48] Tucker, W., A rigorous ode solver and Smale’s 14th problem, Found Comput Math, 2, 53-117 (2002) · Zbl 1047.37012
[49] Young, L., Statistical properties of dynamical systems with some hyperbolicity, Ann Math, 147, 585-650 (1998) · Zbl 0945.37009
[50] Young, L., Recurrence times and rates of mixing, Israel J Math, 110, 153-188 (1999) · Zbl 0983.37005
[51] Young, L., What are SRB measures, and which dynamical systems have them?, J Stat Phys, 108, 733-754 (2002) · Zbl 1124.37307
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