Skip to main content
Log in

Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors

Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We give sufficient conditions for a shift space (Σ, σ) to be intrinsically ergodic, along with sufficient conditions for every subshift factor of Σ to be intrinsically ergodic. As an application, we show that every subshift factor of a β-shift is intrinsically ergodic, which answers an open question included in Mike Boyle’s article “Open problems in symbolic dynamics”. We obtain the same result for S-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.

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

Access this article

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. A. Bertrand-Mathis, Développement en base θ, Bulletin de la Société Mathématique de France 114 (1986), 271–323.

    MathSciNet  MATH  Google Scholar 

  2. F. Blanchard and G. Hansel, Systèmes codés, Theoretical Computer Science 44 (1986), 17–49.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Bowen, Some systems with unique equilibrium states, Mathematical Systems Theory 8 (1974), 193–202.

    Article  MathSciNet  Google Scholar 

  4. M. Boyle, Open problems in symbolic dynamics, Contemporary Mathematics 469 (2008), 69–118.

    Article  MathSciNet  Google Scholar 

  5. A. I. Bufetov and B. M. Gurevich, Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of abelian differentials, Matematicheskiĭ Sbornik 202 (2011), no. 7, 3–42.

    Article  MathSciNet  Google Scholar 

  6. J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel Journal of Mathematics 100 (1997), 125–161.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Buzzi, Subshifts of quasi-finite type, Inventiones Mathematicae 159 (2005), 369–406.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Buzzi and T. Fisher, Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems, Preprint, arXiv:09033692, 2009.

  9. M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.

    MATH  Google Scholar 

  10. D. Fiebig and U.-R. Fiebig, Covers for coded systems, Contemporary Mathematics 135 (1992), 139–179.

    Article  MathSciNet  Google Scholar 

  11. B. M. Gurevic, Uniqueness of the measure with maximal entropy for symbolic dynamical systems that are close to Markov ones, Soviet Mathematics. Doklady 13 (1972), 569–571.

    MathSciNet  Google Scholar 

  12. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel Journal of Mathematics 34 (1979), 213–237.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II, Israel Journal of Mathematics 38 (1981), 107–115.

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Johnson, Beta-shift dynamical systems and their associated languages, Ph.D. thesis, University of North Carolina at Chapel Hill, 1999.

  15. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Vol. 54, Cambridge University Press, 1995.

  16. D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995.

  17. B. Maia, An equivalent system for studying periodic points of the β-transformation for a Pisot or a Salem number, Ph.D. thesis, University of Warwick, 2007, http://www.warwick.ac.uk/~marcq/bmaia thesis.pdf.

  18. W. Parry, On the β-expansions of real numbers, Acta Mathematica Hungarica 11 (1960), 401–416.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Parry, Intrinsic Markov chains, Transactions of the American Mathematical Society 112 (1964), 55–66.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Petersen, Chains, entropy, coding, Ergodic Theory and Dynamical Systems 6 (1986), 415–448.

    Article  MathSciNet  MATH  Google Scholar 

  21. C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory and Dynamical Systems 27 (2007), 929–956.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory and Dynamical Systems 17 (1997), 675–694.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Thomsen, On the structure of beta shifts, Contemporary Mathematics 385 (2005), 321–332.

    Article  MathSciNet  Google Scholar 

  24. D. J. Thompson, Irregular sets, the β-transformation and the almost specification property, Transactions of the American Mathematical Society, to appear. arXiv:0905.0739.

  25. P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, Journal of Statistical Physics 146 (2012), 330–358.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Walters, Equilibrium states for β-transformations and related transformations, Mathematische Zeitschrift 159 (1978), 65–88.

    Article  MathSciNet  Google Scholar 

  27. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer, New York, 1982.

    Google Scholar 

  28. B. Weiss, Intrinsically ergodic systems, American Mathematical Society. Bulletin 76 (1970), 1266–1269.

    Article  MATH  Google Scholar 

  29. B. Weiss, Subshifts of finite type and sofic systems, Monatshefte für Mathematik 77 (1973), 462–474.

    Article  MATH  Google Scholar 

  30. K. Yamamoto, On the weaker forms of the specification property and their applications, Proceedings of the American Mathematical Society 137 (2009), 3807–3814.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vaughn Climenhaga.

Additional information

D. T. is supported by NSF grant DMS-1101576.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Climenhaga, V., Thompson, D.J. Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors. Isr. J. Math. 192, 785–817 (2012). https://doi.org/10.1007/s11856-012-0052-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0052-x

Keywords

Navigation