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Equilibrium states for Mañé diffeomorphisms

Published online by Cambridge University Press:  18 January 2018

VAUGHN CLIMENHAGA ,
TODD FISHER  and
DANIEL J. THOMPSON
VAUGHN CLIMENHAGA
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA email climenha@math.uh.edu
TODD FISHER
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA email tfisher@mathematics.byu.edu
DANIEL J. THOMPSON
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA email thompson@math.osu.edu
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Abstract

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 9 , September 2019 , pp. 2433 - 2455
Copyright
© Cambridge University Press, 2018 

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References

Alves, J. F. and Li, X.. Gibbs–Markov–Young structures with (stretched) exponential tail for partially hyperbolic attractors. Adv. Math. 279 (2015), 405–437.Google Scholar
Arbieto, A. and Prudente, L.. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete Contin. Dyn. Syst. 32(1) (2012), 27–40.Google Scholar
Barreira, L. and Pesin, Y.. Dynamics of systems with nonzero Lyapunov exponents. Nonuniform Hyperbolicity (Encyclopedia of Mathematics and its Applications, 115) . Cambridge University Press, Cambridge, 2007.Google Scholar
Berg, K.. Convolution of invariant measures, maximal entropy. Math. Systems Theory 3 (1969), 146–150.Google Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.Google Scholar
Bowen, R.. Entropy-expansive maps. Trans. Amer. Math. Soc. 164 (1972), 323–331.Google Scholar
Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.Google Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Systems Theory 8(3) (1974–1975), 193–202.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . revised edn. Springer, Berlin, 2008.Google Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.Google Scholar
Bruin, H. and Todd, M.. Equilibrium states for interval maps: the potential - tlog|Df|. Ann. Sci. Éc. Norm. Supér. (4) 42(4) (2009), 559–600.Google Scholar
Buzzi, J., Fisher, T., Sambarino, M. and Vásquez, C.. Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Th. & Dynam. Sys. 32(1) (2012), 63–79.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.. Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Israel J. Math. 130 (2002), 29–75.Google Scholar
Climenhaga, V.. The thermodynamic approach to multifractal analysis. Ergod. Th. & Dynam. Sys. 34(5) (2014), 1409–1450.Google Scholar
Climenhaga, V., Fisher, T. and Thompson, D. J.. Unique equilibrium states for Bonatti–Viana diffeomorphisms. Preprint, 2017, arXiv:1505.06371.Google Scholar
Climenhaga, V. and Thompson, D. J.. Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Adv. Math. 303 (2016), 745–799.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
Crisostomo, J. and Tahzibi, A.. Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part. Preprint, 2016, arXiv:1604.07220.Google Scholar
Denker, M. and Urbański, M.. Ergodic theory of equilibrium states for rational maps. Nonlinearity 4(1) (1991), 103–134.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, New York, 1977.Google Scholar
Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Student Texts, 42) . Cambridge University Press, Cambridge, 1998.Google Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122(3) (1985), 509–539.Google Scholar
Leplaideur, R., Oliveira, K. and Rios, I.. Equilibrium states for partially hyperbolic horseshoes. Ergod. Th. & Dynam. Sys. 31(1) (2011), 179–195.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.Google Scholar
Misiurewicz, M.. Topological conditional entropy. Studia Math. 55(2) (1976), 175–200.Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems (Chicago Lectures in Mathematics) . University of Chicago Press, Chicago, IL, 1997.Google Scholar
Pesin, Y.. Lectures on Partial Hyperbolicity and Stable Ergodicity (Zurich Lectures in Advanced Mathematics) . European Mathematical Society (EMS), Zürich, 2004.Google Scholar
Pesin, Y., Senti, S. and Zhang, K.. Thermodynamics of towers of hyperbolic type. Trans. Amer. Math. Soc. 368(12) (2016), 8519–8552.Google Scholar
Pfister, C.-E. and Sullivan, W. G.. Large deviations estimates for dynamical systems without the specification property. Applications to the 𝛽-shifts. Nonlinearity 18(1) (2005), 237–261.Google Scholar
Pilyugin, S.. Shadowing in Dynamical Systems (Lecture Notes in Mathematics, 1706) . Springer, Berlin, 1999.Google Scholar
Pliss, V. A.. On a conjecture of Smale. Differ. Uravn. 8 (1972), 268–282.Google Scholar
Pujals, E. and Sambarino, M.. A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26(1) (2006), 281–289.Google Scholar
Ramos, V. and Siqueira, J.. On equilibrium states for partially hyperbolic horseshoes: uniqueness and statistical properties. Bull. Braz. Math. Soc. (N.S.) 48(3) (2017), 347–375.Google Scholar
Rios, I. and Siqueira, J.. On equilibrium states for partially hyperbolic horseshoes. Ergod. Th. & Dynam. Sys. First View article, arXiv:1505.07742, 2016.Google Scholar
Ruelle, D.. A measure associated with Axiom-A attractors. Amer. J. Math. 98(3) (1976), 619–654.Google Scholar
Sarig, O.. Phase transitions for countable Markov shifts. Comm. Math. Phys. 217(3) (2001), 555–577.Google Scholar
Senti, S. and Takahasi, H.. Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties. Ergod. Th. & Dynam. Sys. 36(1) (2016), 215–255.Google Scholar
Sinaĭ, J. G.. Gibbs measures in ergodic theory. Uspekhi Mat. Nauk 27(4(166)) (1972), 21–64.Google Scholar
Spatzier, R. and Visscher, D.. Equilibrium measures for certain isometric extensions of Anosov systems. Ergod. Th. & Dynam. Sys. (2016), to appear.Google Scholar
Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Amer. Math. Soc. 140(6) (2012), 1973–1985.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, Berlin, 1982.Google Scholar
Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.Google Scholar