Israel Journal of Mathematics

, Volume 213, Issue 1, pp 475–503 | Cite as

On entropy of dynamical systems with almost specification

  • Dominik KwietniakEmail author
  • Piotr Oprocha
  • Michał Rams


We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.


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  1. [1]
    F. Blanchard, Fully positive topological entropy and topological mixing, in Symbolic Dynamics and its Applications (New Haven, CT, 1991), Contemporary Mathematics, Vol. 135, American Mathematical Society, Providence, RI, 1992, pp. 95–105.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. S. Block and W. A. Coppel, One-Dimensional Dynamics, Lecture Notes in Mathematics, Vol. 1513, Springer-Verlag, Berlin, 1992.Google Scholar
  3. [3]
    R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Transactions of the American Mathematical Society 154 (1971), 377–397.MathSciNetzbMATHGoogle Scholar
  4. [4]
    V. Climenhaga and D. Thompson, Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors, Israel Journal of Mathematics 192 (2012), 785–817.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Dateyama, Invariant measures for homeomorphisms with almost weak specification in Probability Theory and Mathematical Statistics (Tbilisi, 1982), Lecture Notes in Mathematics, Vol. 1021, Springer, Berlin, 1983, pp. 93–96.MathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Dateyama, The almost weak specification property for ergodic group automorphisms of abelian groups, Journal of the Mathematical Society of Japan 42 (1990), 341–351.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003.Google Scholar
  8. [8]
    E. Glasner and B. Weiss, Topological entropy of extensions, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Mathematical Society Lecture Note Series, Vol. 205, Cambridge University Press, Cambridge, 1995, pp. 299–307.MathSciNetGoogle Scholar
  9. [9]
    W. Huang and X. Ye, A local variational relation and applications, Israel Journal of Mathematics 151 (2006), 237–279.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Kulczycki, D. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fundamenta Mathematicae 224 (2014), 241–278.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    P. Kurka, Topological and Symbolic Dynamics, Cours Spécialisés, Vol. 11, Société Mathématique de France, Paris, 2003.Google Scholar
  12. [12]
    D. Kwietniak, M. Łącka and P. Oprocha, A panorama of specificationlike properties and their consequences, Contemporary Mathematics, to appear, arXiv:1503.07355[math.DS], Scholar
  13. [13]
    D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.CrossRefzbMATHGoogle Scholar
  14. [14]
    B. Marcus, A note on periodic points for ergodic toral automorphisms, Monatshefte für Mathematik 89 (1980), 121–129.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Mathematica 67 (1980), 45–63MathSciNetzbMATHGoogle Scholar
  16. [16]
    T. K. S. Moothathu and P. Oprocha, Syndetic proximality and scramed sets, Topological Methods in Nonlinear Analysis 41 (2013), 421–461.MathSciNetzbMATHGoogle Scholar
  17. [17]
    R. Pavlov, On weakenings of the specification property and intrinsic ergodicity, Advances in Mathematics, to appear.Google Scholar
  18. [18]
    C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the β-shifts, Nonlinearity 18 (2005), 237–261.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Thompson, Irregular sets, the beta-transformation and the almost specification property, Transactions of the American Mathematical Society 364 (2012), 5395–5414.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    K. Thomsen, On the ergodic theory of synchronized systems, Ergodic Theory and Dynamical Systems, 26 (2006), 1235–1256.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York–Berlin, 1982.Google Scholar
  22. [22]
    X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, to appear.Google Scholar

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© Hebrew University of Jerusalem 2016

Authors and Affiliations

  • Dominik Kwietniak
    • 1
    Email author
  • Piotr Oprocha
    • 2
    • 3
  • Michał Rams
    • 4
  1. 1.Faculty of Mathematics and Computer ScienceJagiellonian University in KrakówKrakówPoland
  2. 2.AGH University of Science and TechnologyFaculty of Applied MathematicsKrakówPoland
  3. 3.National Supercomputing Centre IT4InnovationsDivision of the University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22OstravaCzech Republic
  4. 4.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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