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Measures of maximal entropy for suspension flows over the full shift

  • Tamara KucherenkoEmail author
  • Daniel J. Thompson
Article
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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.

Mathematics Subject Classification

37D35 37B10 37A35 

Notes

References

  1. 1.
    Bowen, R., Walters, P.: Expansive one-parameter flows. J. Differ. Equ. 12, 180–193 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Climenhaga, V., Pavlov, R.: One-sided almost specification and intrinsic ergodicity. Ergodic Theory Dyn Syst. (2018) [(to appear)Published online as Firstview article]Google Scholar
  5. 5.
    Climenhaga, V., Thompson, D.J.: Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Adv. Math. 303, 744–799 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 7.
    Gelfert, K., Ruggiero, R.: Geodesic flows modeled by expansive flows. Proc. Edinburgh Math. Soc. 62(1), 61–95 (2019)Google Scholar
  8. 8.
    Haydn, N.T.A.: Phase transitions in one-dimensional subshifts. Discret. Contin. Dyn. Syst. 33(5), 1965–1973 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hofbauer, F.: Examples for the nonuniqueness of the equilibrium state. Trans. Am. Math. Soc. 228, 223–241 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Iommi, G., Jordan, T.: Phase transitions for suspension flows. Comm. Math. Phys. 320(2), 475–498 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Iommi, G., Jordan, T., Todd, M.: Recurrence and transience for suspension flows. Israel J. Math. 209(2), 547–592 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Krieger, W: On the uniqueness of the equilibrium state. Math. Syst. Theory 8(2), 97–104 (1974/75)Google Scholar
  13. 13.
    Kwietniak, D., Oprocha, P., Rams, M.: On entropy of dynamical systems with almost specification. Israel J. Math. 213(1), 475–503 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Markley, N.G., Paul, M.E.: Equilibrium states of grid functions. Trans. Am. Math. Soc. 274(1), 169–191 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Number 187-188 in Astérisque. Soc. Math. France (1990)Google Scholar
  17. 17.
    Pavlov, R.: On intrinsic ergodicity and weakenings of the specification property. Adv. Math. 295, 250–270 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Petersen, K.: Chains, entropy, coding. Ergodic Theory Dyn. Syst. 6(3), 415–448 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Savchenko, S.V.: Special flows constructed from countable topological Markov chains. Funktsional. Anal. i Prilozhen. 32(1), 40–53, 96 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Savchenko, S.V.: Equilibrium states with incomplete supports and periodic trajectories. Math. Notes 59(2), 163–179 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ures, R.: Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Am. Math. Soc. 140(6), 1973–1985 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe City College of New YorkNew YorkUSA
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA

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