Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T16:30:21.303Z Has data issue: false hasContentIssue false
  • English
  • Français

SRB measures for partially hyperbolic attractors of local diffeomorphisms

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  17 October 2018

ANDERSON CRUZ  and
PAULO VARANDAS
ANDERSON CRUZ
Affiliation:
Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Av. Rui Barbosa, s/n, 44380-000 Cruz das Almas, BA, Brazil email [email protected]
PAULO VARANDAS
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

Keywords

SRB measurespartially hyperbolicitycone-hyperbolicitystatistical stability

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1545 - 1593
Copyright
© Cambridge University Press, 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2) (2000), 351–398.Google Scholar
Alves, J. F., Carvalho, M. and Freitas, J.. Statistical stability and continuity of SRB entropy for systems with Gibbs–Markov structures. Comm. Math. Phys. 296(3) (2010), 739–767.Google Scholar
Alves, J. F., Luzzatto, S. and Pinheiro, V.. Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 22(6) (2005), 817–839.Google Scholar
Aoki, N. and Hiraide, K.. Topological Theory of Dynamical Systems: Recent Advances. Vol. 52. Elsevier, Amsterdam, 1994.Google Scholar
Baladi, V.. Linear response, or else. Proceedings of the ICM 2014, Seoul. Vol. III. pp. 525–545; http://www.icm2014.org/en/vod/proceedings.html.Google Scholar
Barreira, L. and Pesin, Y.. Non Uniform Hyperbolicity: Dynamics of Systems With Non Zero Lyapunov Exponents. Cambridge University Press, New York, 2007.Google Scholar
Blank, M., Keller, G. and Liverani, C.. Ruelle-Perron-Frobenius spectrum for Anosov maps. Nonlinearity 15(6) (2002), 1905–1973.Google Scholar
Bomfim, T., Castro, A. and Varandas, P.. Differentiability of thermodynamical quantities in non-uniformly expanding dynamics. Adv. Math. 292 (2016), 478–528.Google Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115(1) (2000), 157–193.Google Scholar
Bortolotti, R.. Physical measures for certain partially hyperbolic attractors on 3-manifolds. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2017.24. Published online 8 May 2017.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.Google Scholar
Carvalho, M.. Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 13(1) (1993), 21–44.Google Scholar
Carvalho, M. and Varandas, P.. (Semi)continuity of the entropy of Sinai probability measures for partially hyperbolic diffeomorphisms. J. Math. Anal. Appl. 434(2) (2016), 1123–1137.Google Scholar
Castro, A.. Fast mixing for attractors with mostly contracting central direction. Ergod. Th. & Dynam. Sys. 24 (2004), 17–44.Google Scholar
Castro, A. and Varandas, P.. Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 30(2) (2013), 225–249.Google Scholar
Cowieson, W. and Young, L.-S.. SRB measures as zero-noise limits. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1115–1138.Google Scholar
Demers, M. and Liverani, C.. Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9) (2008), 4777–4814.Google Scholar
Dolgopyat, D.. On dynamics of mostly contracting diffeomorphisms. Comm. Math. Phys. 213(1) (2000), 181–201.Google Scholar
Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189–217.Google Scholar
Iglesias, J., Lizana, C. and Portela, A.. Robust transitivity for endomorphisms admitting critical points. Proc. Amer. Math. Soc. 144(3) (2016), 1235–1250.Google Scholar
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) (1985), 509–539.Google Scholar
Liu, P.-D.. Pesin’s entropy formula for endomorphisms. Nagoya Math. J. 150 (1998), 197–209.Google Scholar
Liu, P.-D.. Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 240 (2003), 531–538.Google Scholar
Liu, P.-D. and Shu, L.. Absolute continuity of hyperbolic invariant measures for endomorphisms. Nonlinearity 24 (2011), 1595–1611.Google Scholar
Liverani, C.. Decay of correlations. Ann. of Math. (2) 142 (1995), 239–301.Google Scholar
Mané, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.Google Scholar
Mehdipour, P.. On the uniqueness of SRB measures for endomorphisms. Preprint, 2017, arXiv:1703. 06332v1 [math.DS].Google Scholar
Mehdipour, P. and Tahzibi, A.. SRB measures and homoclinic relation for endomorphisms. J. Stat. Phys. 163(1) (2016), 139–155.Google Scholar
Mi, Z., Cao, Y. and Yang, D.. SRB measures for attractors with continuous invariant splittings. Math. Z. (2017), doi:10.1007/s00209-017-1883-2.Google Scholar
Mihailescu, E.. Physical measures for multivalued inverse iterates near hyperbolic repellors. J. Stat. Phys. 139(5) (2010), 800–819.Google Scholar
Mihailescu, E. and Urbanski, M.. Entropy production for a class of inverse SRB measures. J. Stat. Phys. 150(5) (2013), 881–888.Google Scholar
Mora, L. and Viana, M.. Abundance of strange attractors. Acta Math. 171(1) (1993), 1–71.Google Scholar
Pesin, Y. and Sinai, Y.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 417–438.Google Scholar
Przytycki, F.. Anosov endomorphisms. Studia Math. 3(58) (1976), 249–285.Google Scholar
Pugh, C. and Shub, M.. Ergodic attractors. Trans. Amer. Math. Soc. 312(1) (1989), 1–54.Google Scholar
Qian, M., Xie, J.-S. and Zhu, S.. Smooth Ergodic Theory for Endomorphisms. Springer, Berlin, 2009.Google Scholar
Qian, M. and Zhang, Z.. Ergodic theory for Axiom A endomorphisms. Ergod. Th. & Dynam. Sys. 15(1) (1995), 161–174.Google Scholar
Qian, M. and Zhu, S.. SRB measures and Pesin’s entropy formula for endomorphisms. Trans. Amer. Math. Soc. 354(4) (2002), 1453–1471.Google Scholar
Rokhlin, V.. On the fundamental ideas of measure theory. Matematicheskii Sbornik 67(1) (1949), 107–150.Google Scholar
Ruelle, D.. A measure associated with Axiom-A attractors. Amer. J. Math. (1976), 619–654.Google Scholar
Shu, L.. The metric entropy of endomorphisms. Commun. Math. Phys. 291 (2009), 491–512.Google Scholar
Shub, M.. Global Stability of Dynamical Systems. Springer, Berlin, 1987.Google Scholar
Sinai, Y.. Gibbs measures in ergodic theory. Russian Math. Surveys 27(4) (1972), 21–69.Google Scholar
Sumi, N.. A class of differentiable total maps which are topologically mixing. Proc. Amer. Math. Soc. 127(3) (1999), 915–924.Google Scholar
Tsujii, M.. Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1) (2005), 37–132.Google Scholar
Urbánski, M. and Wolf, C.. SRB measures for Axiom A endomorphisms. Math. Res. Lett 11(5-6) (2004), 785–797.Google Scholar
Ures, R.. On the approximation of Hénon like attractors by homoclinic tangencies. Ergod. Th. & Dynam. Sys. 15 (1995), 1223–1229.Google Scholar
Varandas, P. and Viana, M.. Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 27(2) (2010), 555–593.Google Scholar
Vásquez, C.. Statistical stability for diffeomorphisms with dominated splitting. Ergod. Th. & Dynam. Sys. 27(01) (2007), 253–283.Google Scholar
Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 558–650.Google Scholar