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Bernoulli Property of Equilibrium States for Certain Partially Hyperbolic Diffeomorphisms

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Abstract

Let \(f:M\rightarrow M\) be a \(C^1\) partially hyperbolic diffeomorphism which is topologically transitive and Lyapunov stable, and \(\varphi :M\rightarrow {\mathbb R}\) a continuous function satisfying the Bowen properties. Then there exists a unique equilibrium state for \((M,f,\varphi )\) by Climenhaga et al. (J Mod Dyn 16:155–205, 2020). In this paper, we show that if f is topologically mixing, then the unique equilibrium state has the Bernoulli property. We also investigate the Bernoulli property for the volume measure, as well as u-equilibrium states.

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References

  1. Bonatti, C., Díaz, L.J., Viana, M.: Dynamics beyond uniform hyperbolicity. A global geometric and probabilitistic perspective. In: Encyclopaedia Math. Sci., vol. 102. Springer-Verlag (2005)

  2. Bonatti, C., Zhang, J.: Transitive Partially Hyperbolic Diffeomorphisms with One-Dimensional Neutral Center. arXiv preprint arXiv:1904.05295 (2019)

  3. Bowen, R.: Some systems with unique equilibrium states. Math. Syst. Theory 8(3), 193–202 (1974)

    MathSciNet  MATH  Google Scholar 

  4. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Math., vol. 470. Springer Verlag, Berlin (1975)

    Google Scholar 

  5. Burns, K., Climenhaga, V., Fisher, T., Thompson, D.J.: Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28, 1209–1259 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Burns, K., Pollicott, M.: Stable ergodicity and frame flows. Geom. Dedic. 98(1), 189–210 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Burns, K., Wilkinson, A.: On the ergodicity of partially hyperbolic systems. Ann. Math. 451–489 (2010)

  8. Buzzi, J., Fisher, T., Sambarino, M., Vásquez, C.: Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Theory Dyn. Syst. 32(1), 63–79 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Call, B., Thompson, D.J.: Equilibrium States for Products of Flows and the Mixing Properties of Rank 1 Geodesic Flows. arXiv preprint arXiv:1906.09315 (2019)

  10. Chernov, N.I., Haskell, C.: Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Theory Dyn. Syst. 16(1), 19–44 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Climenhaga, V., Fisher, T., Thompson, D.J.: Unique equilibrium states for Bonatti–Viana diffeomorphisms. Nonlinearity 31(6), 2532–2570 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Climenhaga, V., Fisher, T., Thompson, D.J.: Equilibrium states for Mañé diffeomorphisms. Ergod. Theory Dyn. Syst. 39(9), 2433–2455 (2019)

    MATH  Google Scholar 

  13. Climenhaga, V., Pesin, Y.B., Zelerowicz, A.: Equilibrium states in dynamical systems via geometric measure theory. Bull. Am. Math. Soc. 56(4), 569–610 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Climenhaga, V., Pesin, Y.B., Zelerowicz, A.: Equilibrium measures for some partially hyperbolic systems. J. Mod. Dyn. 16, 155–205 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Climenhaga, V., Thompson, D.J.: Intrinsic ergodicity via obstruction entropies. Ergod. Theory Dyn. Syst. 34(6), 1816–1831 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Climenhaga, V., Thompson, D.J.: Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Adv. Math. 303, 745–799 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Crisostomo, J., Tahzibi, A.: Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part. Nonlinearity 32(2), 584–602 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Dani, S.G.: Dynamical systems on homogeneous spaces. Bull. Am. Math. Soc. 82(6), 950–952 (1976)

    MathSciNet  MATH  Google Scholar 

  19. Dani, S.G.: Bernoulli translations and minimal horospheres on homogeneous spaces. J. Indian Math. Soc. 39, 245–284 (1976)

    MATH  Google Scholar 

  20. Díaz, L.J., Gelfert, K., Rams, M.: Rich phase transitions in step skew products. Nonlinearity 24(12), 3391–3412 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Díaz, L.J., Gelfert, K., Rams, M.: Abundant rich phase transitions in step-skew products. Nonlinearity 27(9), 2255–2280 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Dolgopyat, D., Kanigowski, A., Rodriguez-Hertz, F.: Exponential Mixing Implies Bernoulli. arXiv preprint arXiv:2106.03147 (2021)

  23. Dong, C., Kanigowski, A.: Bernoulli Property for Certain Skew Products Over Hyperbolic Systems. arXiv preprint arXiv:1912.08132 (2019)

  24. Franco, E.: Flows with unique equilibrium states. Am. J. Math. 99(3), 486–514 (1977)

    MathSciNet  MATH  Google Scholar 

  25. Glasner, E.: Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, vol. 101, xii+384 pp. American Mathematical Society (2003)

  26. Hu, H., Hua, Y., Wu, W.: Unstable entropies and variational principle for partially hyperbolic diffeomorphisms. Adv. Math. 321, 31–68 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Hu, H., Wu, W., Zhu, Y.: Unstable pressure and \(u\)-equilibrium states for partially hyperbolic diffeomorphisms. Ergod. Theory Dyn. Syst. (to appear)

  28. Kanigowski, A.: Bernoulli Property for Homogeneous Systems. arXiv preprint arXiv:1812.03209 (2018)

  29. Kanigowski, A., Rodriguez Hertz, F., Vinhage, K.: On the non-equivalence of the Bernoulli and K properties in dimension four. J. Mod. Dyn. 13, 221–250 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Katok, A.: Smooth non-Bernoulli K-automorphisms. Invent. Math. 61, 291–300 (1980)

    MathSciNet  MATH  Google Scholar 

  31. Katzenlson, Y.: Ergodic automorphisms of \(T^n\) are Bernoulli shifts. Israel J. Math. 10(2), 186–195 (1971)

    MathSciNet  Google Scholar 

  32. Knieper, G.: The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Ann. Math. 148, 291–314 (1998)

    MathSciNet  MATH  Google Scholar 

  33. Ledrappier, F.: Mesures déquilibre dentropie complètement positive. Astérisque 50, 251–272 (1977)

    MathSciNet  Google Scholar 

  34. Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms: part II: relations between entropy, exponents and dimension. Ann. Math. 122, 540–574 (1985)

    MathSciNet  MATH  Google Scholar 

  35. Margulis, G.A.: On Some Aspects of the Theory of Anosov Systems. With a Survey By Richard Sharp: Periodic Orbits of Hyperbolic Flows. Springer Monographs in Mathematics, Springer-Verlag, Berlin (2004)

    MATH  Google Scholar 

  36. Ornstein, D.: Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4, 337–352 (1970)

    MathSciNet  MATH  Google Scholar 

  37. Ornstein, D.: An example of a Kolmogorov automorphism that is not a Bernoulli shift. Adv. Math. 10, 49–62 (1973)

    MathSciNet  MATH  Google Scholar 

  38. Ornstein, D., Weiss, B.: Geodesic flows are Bernoullian. Israel J. Math. 14, 184–198 (1973)

    MathSciNet  MATH  Google Scholar 

  39. Ornstein, D., Weiss, B.: On the Bernoulli nature of systems with some hyperbolic structure. Ergod. Theory Dyn. Syst. 18(2), 441–456 (1998)

    MathSciNet  MATH  Google Scholar 

  40. Pesin, Y.B.: Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32(4), 55–114 (1977)

    MATH  Google Scholar 

  41. Pesin, Y.B.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2008)

    Google Scholar 

  42. Ponce, G.: Lyapunov Stability and the Bernoulli Property. arXiv preprint arXiv:1906.05396 (2019)

  43. Ponce, G., Tahzibi, A., Varão, R.: On the Bernoulli property for certain partially hyperbolic diffeomorphisms. Adv. Math. 329, 329–360 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Ratner, M.: Anosov flows with Gibbs measures are also Bernoullian. Israel J. Math. 17(4), 380–391 (1974)

    MathSciNet  MATH  Google Scholar 

  45. Rodriguez Hertz, F., Rodriguez Hertz, M.A., Ures, R.: A survey of partially hyperbolic dynamics. Fields Inst. Commun. 51, 35–88 (2007)

    MathSciNet  MATH  Google Scholar 

  46. Rodriguez Hertz, F., Rodriguez Hertz, M.A., Tahzibi, A., Ures, R.: Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Theory Dyn. Syst. 32(2), 825–839 (2012)

    MathSciNet  MATH  Google Scholar 

  47. Rohlin, V.A.: On the fundamental ideas of measure theory. J Am. Math. Soc. Transl. 1952(71), 55 (1952)

    MathSciNet  Google Scholar 

  48. Rohlin, V.A., Sinai, J.G.: Construction and properties of invariant measurable partitions. Dokl. Akad. Nauk. SSSR 141, 1038–1041 (1961)

    MathSciNet  Google Scholar 

  49. Ruelle, D.: Statistical mechanics on a compact set with \(Z^{v}\) action satisfying expansiveness and specification. Trans. Am. Math. Soc. 185, 237–251 (1973)

    MATH  Google Scholar 

  50. Ruelle, D.: Thermodynamical formalism, the mathematical structures of classical equilibrium statistical mechanics. In: Encyclopedia Math. Appl., vol. 5. Addison-Wesley Publishing Co, Reading (1978)

  51. Spatzier, R., Visscher, D.: Equilibrium measures for certain isometric extensions of Anosov systems. Ergod. Theory Dyn. Syst. 38(3), 1154–1167 (2018)

    MathSciNet  MATH  Google Scholar 

  52. Walters, P.: A variational principle for the pressure of continuous transformations. Am. J. Math. 97(4), 937–971 (1975)

    MathSciNet  MATH  Google Scholar 

  53. Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer Science & Business Media, Berlin (2000)

    Google Scholar 

  54. Yang, J.: Entropy Along Expanding Foliations. arXiv:1601.05504

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Acknowledgements

This work is supported by NSFC Nos. 12071474 and 11701559. The authors would like to thank the referees for valuable comments and suggestions.

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  1. Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China

    Xiaochen Li

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Weisheng Wu

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  1. Xiaochen Li
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Li, X., Wu, W. Bernoulli Property of Equilibrium States for Certain Partially Hyperbolic Diffeomorphisms. J Dyn Diff Equat 35, 1843–1862 (2023). https://doi.org/10.1007/s10884-021-10057-7

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  • DOI: https://doi.org/10.1007/s10884-021-10057-7

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