Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-29T19:05:21.805Z Has data issue: false hasContentIssue false
  • English
  • Français

The K-property for some unique equilibrium states in flows and homeomorphisms

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

Published online by Cambridge University Press:  12 November 2021

BENJAMIN CALL Open the ORCID record for BENJAMIN CALL [Opens in a new window]
BENJAMIN CALL*
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH43210, USA
*
e-mail: call.119@buckeyemail.osu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.

Keywords

equilibrium statesthermodynamic formalismKolmogorov property

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37C40: Smooth ergodic theory, invariant measures 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2509 - 2532
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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

Abramov, L. M. and Rohlin, V. A.. Entropy of a skew product of mappings with invariant measure. Vestnik Leningrad. Univ. 17(7) (1962) 5–13.Google Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Syst. Theory 8(3) (1974/75), 193–202.CrossRefGoogle Scholar
Burns, K., Climenhaga, V., Fisher, T. and Thompson, D. J.. Unique equilibrium states for geodesic flows in nonpositive curvature . Geom. Funct. Anal. 28(5) (2018), 1209–1259.CrossRefGoogle Scholar
Buzzi, J., Fisher, T., Sambarino, M. and Vásquez, C.. Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Th. & Dynam. Syst. 32(1) (2012), 63–79.CrossRefGoogle Scholar
Call, B. and Thompson, D. J.. Equilibrium states for products of flows and the mixing properties of rank 1 geodesic flows. Preprint, 2019, arXiv:1906.09315.Google Scholar
Chen, D., Kao, L.-Y. and Park, K.. Unique equilibrium states for geodesic flows over surfaces without focal points. Nonlinearity 33(3) (2020), 1118–1155.CrossRefGoogle Scholar
Chernov, N. I. and Haskell, C.. Nonuniformly hyperbolic $K$ -systems are Bernoulli Ergod. Th. & Dynam. Syst. 16(1) (1996), 19–44.CrossRefGoogle Scholar
Climenhaga, V., Fisher, T. and Thompson, D. J.. Equilibrium states for Mañé diffeomorphisms. Ergod. Th. & Dynam. Syst. 33(9) (2018), 1–23.Google Scholar
Climenhaga, V., Fisher, T. and Thompson, D. J. Unique equilibrium states for Bonatti–Viana diffeomorphisms. Nonlinearity 31(6) (2018), 2532.CrossRefGoogle Scholar
Climenhaga, V. and Pesin, Y.. Building thermodynamics for non-uniformly hyperbolic maps. Arnold Math. J. 3(1) (2017), 37–82.CrossRefGoogle Scholar
Climenhaga, V. and Thompson, D. J.. Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Adv. Math. 303 (2016), 745–799.CrossRefGoogle Scholar
Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Y. G.. Ergodic Theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245). Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ.CrossRefGoogle Scholar
Denker, M., Grillenberger, C., and Sigmund, K.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer-Verlag, Berlin, 1976.CrossRefGoogle Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Downarowicz, T. and Serafin, J.. Fiber entropy and conditional variational principles in compact non-metrizable spaces. Fund. Math. 172(3) (2002), 217–247.CrossRefGoogle Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory (M. B. Porter Lectures). Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
Kanigowski, A., Hertz, F. R. and Vinhage, K.. On the non-equivalence of the Bernoulli and $K$ properties in dimension four. J. Mod. Dyn. 13 (2018), 221–250.CrossRefGoogle Scholar
Katok, A.. Bernoulli diffeomorphisms on surfaces. Ann. Math. (2) 110(3) (1979), 529–547.CrossRefGoogle Scholar
Ledrappier, F. . Mesures d’équilibre d’entropie complètement positive. Astérisque 50 (1977), 251–272.Google Scholar
Ledrappier, F.. A variational principle for the topological conditional entropy. Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) (Lecture Notes in Mathematics, 729). Springer, Berlin, 1979, pp. 78–88.CrossRefGoogle Scholar
Ledrappier, F., Lima, Y. and Sarig, O.. Ergodic properties of equilibrium measures for smooth three dimensional flows. Comment. Math. Helv. 91(1) (2016), 65–106.CrossRefGoogle Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.CrossRefGoogle Scholar
Ornstein, D. and Weiss, B.. On the Bernoulli nature of systems with some hyperbolic structure. Ergod. Th. & Dynam. Syst. 18(2) (1998), 441–456.CrossRefGoogle Scholar
Pesin, Y.B.. Dimension Theory in Dynamical Systems (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, 1997.CrossRefGoogle Scholar
Pilyugin, S. Y.. Shadowing in Dynamical Systems (Lecture Notes in Mathematics, 1706). Springer-Verlag, Berlin, 1999.Google Scholar
Ponce, G., Tahzibi, A. and Varão, R.. On the Bernoulli property for certain partially hyperbolic diffeomorphisms. Adv. Math. 329 (2018), 329–360.CrossRefGoogle Scholar
Ratner, M.. Anosov flows with Gibbs measures are also Bernoullian. Israel J. Math. 17 (1974), 380–391.CrossRefGoogle Scholar
Rohlin, V. A.. Exact endomorphisms of a Lebesgue space. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499–530.Google Scholar
Shahidi, F. and Zelerowicz, A.. Thermodynamics via inducing. J. Stat. Phys. 175(2) (2019), 351–383.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
Wang, T.. Unique equilibrium states, large deviations and Lyapunov spectra for the Katok map. Ergod. Th. & Dynam. Syst. 41(7) (2021), 2182–2219.CrossRefGoogle Scholar