Skip to main content
SpringerLink
Log in

Measures of maximal entropy for suspension flows over the full shift

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We consider suspension flows with continuous roof function over the full shift \(\Sigma \) on a finite alphabet. For any positive entropy subshift of finite type \(Y \subset \Sigma \), we explicitly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over \(\Sigma \) are exactly the lifts of the measure(s) of maximal entropy for Y. In the case when Y is transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If Y has more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a Hölder continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bowen, R., Walters, P.: Expansive one-parameter flows. J. Differ. Equ. 12, 180–193 (1972)

    Article  MathSciNet  Google Scholar 

  2. Bufetov, A.I., Gurevich, B.M.: Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials. SB Math. 202(7), 935–970 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Climenhaga, V., Pavlov, R.: One-sided almost specification and intrinsic ergodicity. Ergodic Theory Dyn Syst. (2018) [(to appear)Published online as Firstview article]

  5. Climenhaga, V., Thompson, D.J.: Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Adv. Math. 303, 744–799 (2016)

    Article  MathSciNet  Google Scholar 

  6. Constantine, D., Lafont, J-F., Thompson, D.J.: The weak specification property for geodesic flows on CAT(-1) spaces. Groups Geom. Dyn. arXiv:1606.06253 (2019) (to appear)

  7. Gelfert, K., Ruggiero, R.: Geodesic flows modeled by expansive flows. Proc. Edinburgh Math. Soc. 62(1), 61–95 (2019)

    Article  MathSciNet  Google Scholar 

  8. Haydn, N.T.A.: Phase transitions in one-dimensional subshifts. Discret. Contin. Dyn. Syst. 33(5), 1965–1973 (2013)

    Article  MathSciNet  Google Scholar 

  9. Hofbauer, F.: Examples for the nonuniqueness of the equilibrium state. Trans. Am. Math. Soc. 228, 223–241 (1977)

    Article  MathSciNet  Google Scholar 

  10. Iommi, G., Jordan, T.: Phase transitions for suspension flows. Comm. Math. Phys. 320(2), 475–498 (2013)

    Article  MathSciNet  Google Scholar 

  11. Iommi, G., Jordan, T., Todd, M.: Recurrence and transience for suspension flows. Israel J. Math. 209(2), 547–592 (2015)

    Article  MathSciNet  Google Scholar 

  12. Krieger, W: On the uniqueness of the equilibrium state. Math. Syst. Theory 8(2), 97–104 (1974/75)

    Article  MathSciNet  Google Scholar 

  13. Kwietniak, D., Oprocha, P., Rams, M.: On entropy of dynamical systems with almost specification. Israel J. Math. 213(1), 475–503 (2016)

    Article  MathSciNet  Google Scholar 

  14. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  15. Markley, N.G., Paul, M.E.: Equilibrium states of grid functions. Trans. Am. Math. Soc. 274(1), 169–191 (1982)

    Article  MathSciNet  Google Scholar 

  16. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Number 187-188 in Astérisque. Soc. Math. France (1990)

  17. Pavlov, R.: On intrinsic ergodicity and weakenings of the specification property. Adv. Math. 295, 250–270 (2016)

    Article  MathSciNet  Google Scholar 

  18. Petersen, K.: Chains, entropy, coding. Ergodic Theory Dyn. Syst. 6(3), 415–448 (1986)

    Article  MathSciNet  Google Scholar 

  19. Savchenko, S.V.: Special flows constructed from countable topological Markov chains. Funktsional. Anal. i Prilozhen. 32(1), 40–53, 96 (1998)

    Article  MathSciNet  Google Scholar 

  20. Savchenko, S.V.: Equilibrium states with incomplete supports and periodic trajectories. Math. Notes 59(2), 163–179 (1996)

    Article  MathSciNet  Google Scholar 

  21. Ures, R.: Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Am. Math. Soc. 140(6), 1973–1985 (2012)

    Article  MathSciNet  Google Scholar 

  22. Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The City College of New York, New York, NY, 10031, USA

    Tamara Kucherenko

  2. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Daniel J. Thompson

Authors
  1. Tamara Kucherenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel J. Thompson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tamara Kucherenko.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

T.K. is supported by Grants from the Simons Foundation #430032 and from the PSC-CUNY TRADA-47-18. D.T. is supported by NSF Grant DMS-1461163.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kucherenko, T., Thompson, D.J. Measures of maximal entropy for suspension flows over the full shift. Math. Z. 294, 769–781 (2020). https://doi.org/10.1007/s00209-019-02287-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02287-9

Mathematics Subject Classification

Search

Navigation