Israel Journal of Mathematics

, Volume 100, Issue 1, pp 125–161 | Cite as

Intrinsic ergodicity of smooth interval maps

Article

Abstract

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy.

In this way, we reduce the study ofC r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. M. Blokh,Decomposition of dynamical systems on an interval, Russian Mathematical Surveys38 (1983), 133–134.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. Bowen,Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society153 (1971), 401–414.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. Bowen,Entropy-expansive maps, Transactions of the American Mathematical Society164 (1972), 323–333.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Buzzi,Number of equilibrium states of piecewise monotonics maps of the interval, Proceedings of the American Mathematical Society123 (1995), 2901–2907.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    J. Buzzi,Intrinsic ergodicity of affine maps on [0, 1]d, Monatshefte für Matematik (to appear).Google Scholar
  6. [6]
    M. Denker, C. Grillenberg and K. Sigmund,Ergodic theory on compact spaces, Lecture Notes in Mathematics527, Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  7. [7]
    M. Gromov,Entropy, homology and semi-algebraic geometry, Séminaire Bourbaki663, 1985–1986.Google Scholar
  8. [8]
    B. M. Gurevič,Topological entropy of enumerable Markov chains, Soviet Mathematics Doklady10 (1969), 911–915.Google Scholar
  9. [9]
    B. M. Gurevič,Shift entropy and Markov measures in the path space of a denumerable graph, Soviet Mathematics Doklady11 (1970), 744–747.Google Scholar
  10. [10]
    F. Hofbauer,On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel Journal of Mathematics34 (1979), 213–237;38 (1981), 107–115.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    F. Hofbauer,The structure of piecewise monotonic transformations, Ergodic Theory and Dynamical Systems1 (1981), 159–178.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Katok,Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l’Institut des Hautes Études Scientifiques51 (1980), 137–173.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Misiurewicz,A short proof of the variational principle for a ℤ +n action on a compact space, Astérisque (International Conference on Dynamical Systems in Mathematical Physics), Vol. 40, 1976, pp. 147–158.MathSciNetGoogle Scholar
  14. [14]
    S. E. Newhouse,Continuity properties of the entropy, Annals of Mathematics129 (1989), 215–237.CrossRefMathSciNetGoogle Scholar
  15. [15]
    S. E. Newhouse,On some results of F. Hofbauer on maps of the interval, inDynamical Systems and Related Topics (K. Shiraiwa, ed.), Proc. Nagoya 1990, World Scientific, Singapore, 1991, pp. 407–422.Google Scholar
  16. [16]
    S. E. Newhouse and L.-S. Young,Dynamics of certain skew products, inGeometric Dynamics, Proc. Rio de Janeiro 1981, Lecture Notes in Mathematics1007, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
  17. [17]
    K. Petersen,Ergodic Theory, Cambridge University Press, 1983.Google Scholar
  18. [18]
    I. A. Salama,Topological entropy and classification of countable chains, Ph.D. thesis, University of North Carolina, Chapel Hill, 1984.Google Scholar
  19. [19]
    S. Smale,Differentiable dynamics, Bulletin of the American Mathematical Society73 (1967), 97–116.MathSciNetGoogle Scholar
  20. [20]
    D. Vere-Jones,Geometric ergodicity in denumerable Markov chains, The Quarterly Journal of Mathematics, Oxford13 (1962), 7–28.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    D. Vere-Jones,Ergodic properties of nonnegative matrices I, Pacific Journal of Mathematics22 (1967), 361–386.MATHMathSciNetGoogle Scholar
  22. [22]
    B. Weiss,Intrinsically ergodic systems, Bulletin of the American Mathematical Society76 (1970), 1266–1269.MATHMathSciNetGoogle Scholar
  23. [23]
    Y. Yomdin,Volume growth and entropy, Israel Journal of Mathematics57 (1987), 285–300.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Y. Yomdin,C k-resolution of semi-algebraic mappings, Israel Journal of Mathematics57 (1987), 301–318.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1997

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Paris-SudOrsayFrance

Personalised recommendations