Israel Journal of Mathematics

, Volume 127, Issue 1, pp 253–277 | Cite as

MixingCrmaps of the interval without maximal measure

  • Sylvie Ruette


We construct aCrtransformation of the interval (or the torus) which is topologically mixing but has no invariant measure of maximal entropy. Whereas the assumption ofC ensures existence of maximal measures for an interval map, it shows we cannot weaken the smoothness assumption. We also compute the local entropy of the example.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ll. Alsedà, J. Llibre and M. Misiurewicz,Combinatorial Dynamics and Entropy in Dimension One, World Scientific, River Edge, NJ, 1993.MATHGoogle Scholar
  2. [2]
    L. S. Block and W. A. Coppel,Dynamics in One Dimension, Lecture Notes in Mathematics1513, Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  3. [3]
    A. M. Blokh,On sensitive mappings of the interval, Russian Mathematical Surveys37 (1982), 203–204.MATHCrossRefGoogle Scholar
  4. [4]
    A. M. Blokh,Decomposition of dynamical systems on an interval, Russian Mathematical Surveys38 (1983), 133–134.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. Bowen,Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society153 (1971), 401–414.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Bowen,Periodic points and measures for Axiom A diffeomorphisms, Transactions of the American Mathematical Society154 (1971), 377–397.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    R. Bowen,Some systems with unique equilibrium states, Mathematical Systems Theory8 (1974), 193–202.CrossRefMathSciNetGoogle Scholar
  8. [8]
    J. Buzzi,personal communication.Google Scholar
  9. [9]
    J. Buzzi,Intrinsic ergodicity of smooth interval maps, Israel Journal of Mathematics100 (1997), 125–161.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Buzzi,Specification on the interval, Transactions of the American Mathematical Society349 (1997), 2737–2754.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Denker, C. Grillenberger and K. Sigmund,Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics527, Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  12. [12]
    B. M. Gurevich and A. S. Zargaryan,A continuous one-dimensional mapping without a measure with maximal entropy, Functional Analysis and its Applications20 (1986), 134–136.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    B. M. Gurevič,Topological entropy of enumerable Markov chains, Soviet Mathematics Doklady10 (1969), 911–915.Google Scholar
  14. [14]
    B. M. Gurevič,Shift entropy and Markov measures in the path space of a denumerable graph, Soviet Mathematics Doklady11 (1970), 744–747.Google Scholar
  15. [15]
    F. Hofbauer,On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel Journal of MathematicsI 34 (1979), 213–237;II 38 (1981), 107–115.CrossRefMathSciNetGoogle Scholar
  16. [16]
    F. Hofbauer,Piecewise invertible dynamical systems, Probability Theory and Related Fields72 (1986), 359–386.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Misiurewicz and W. Szlenk,Entropy of piecewise monotone mappings, Studia Mathematica67 (1980), 45–63.MATHMathSciNetGoogle Scholar
  18. [18]
    S. E. Newhouse,Continuity properties of entropy, Annals of Mathematics129 (1989), 215–235.CrossRefMathSciNetGoogle Scholar
  19. [19]
    K. Petersen,Ergodic Theory, Cambridge University Press, 1983.Google Scholar
  20. [20]
    D. Ruelle,Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity5 (1992), 1223–1236.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    I. A. Salama,Topological entropy and classification of countable chains, PhD thesis, University of North Carolina, Chapel Hill, 1984.Google Scholar
  22. [22]
    I. A. Salama,Topological entropy and recurrence of countable chains, Pacific Journal of Mathematics134 (1988), 325–341;Errata, Pacific Journal of Mathematics,140 (1989), 397.MathSciNetGoogle Scholar
  23. [23]
    D. Vere-Jones,Geometric ergodicity in denumerable Markov chains, The Quarterly Journal of Mathematics13 (1962), 7–28.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Y. Yomdin,Volume growth and entropy, Israel Journal of Mathematics57 (1987), 285–300;C k-resolution of semialgebraic mappings. Addendum to “Volume growth and entropy”, Israel Journal of Mathematics57 (1987), 301–318.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2002

Authors and Affiliations

  • Sylvie Ruette
    • 1
  1. 1.Institut de Mathématiques de LuminyCNRS, case 907Marseille cedex 9France

Personalised recommendations