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Continuity of the measure of maximal entropy for unimodal maps on the interval

  • Peter Raith
Article

Abstract

LetT: [0, 1]→[0, 1] be a unimodal map with positive topological entropy. ThenT has a unique measure μ(T) of maximal entropy. It is proved that the mapT↦μ(T) is continuous with respect to the weak star-topology.

Key words

unimodal map perturbation topological entropy measure of maximal entropy Markov diagram 

References

  1. 1.
    F. Hofbauer,Maximal measures for simple piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete52 (1980), 289–300.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    F. Hofbauer,On intrinsic ergodicity of piecewise monotonic transformations with positive entropy 2, Israel J. Math.38 (1981), 107–115.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. Hofbauer,Piecewise invertible dynamical systems, Probab. Theory Related Fields72 (1986), 359–386.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Misiurewicz,Jumps of entropy in one dimension, Fund. Math.132 (1989), 215–226.MATHMathSciNetGoogle Scholar
  5. 5.
    M. Misiurewicz, S. V. Shlyachkov,Entropy of piecewise continuous interval maps, in:European conference on iteration theory (ECIT 89), Batschuns, 1989, pp. 239–245, (eds.:Ch. Mira,N. Netzer,C. Simó,Gy. Targoński), World Scientific, Singapore, (1991).Google Scholar
  6. 6.
    M. Misiurewicz, W. Szlenk,Entropy of piecewise monotone mappings, Studia Math.67 (1980), 45–63.MATHMathSciNetGoogle Scholar
  7. 7.
    P. Raith,Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math.80 (1992), 97–133.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Raith,Stability of the maximal measure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems17 (1997), 1419–1436.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    P. Raith,Stability of the topological pressure for piecewise monotonic maps under C 1-perturbations, J. Anal. Math.78 (1999), 117–142.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Raith,Perturbations of a topologically transitive piecewise monotonic map on the interval, in:Proceedings of the European conference on iteration theory (ECIT 1996), Urbino, 1996, Grazer Math. Ber.339 (1999), pp. 301–312, (eds.:L. Gardini, G. L. Forti, D. Gronau, L. Paganoni).Google Scholar
  11. 11.
    P. Raith,Discontinuities of the pressure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems21 (2001), 197–232.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    P. Walters,An introduction to ergodic theory, Graduate Texts in Mathematics 79, Springer, New York, (1982).MATHGoogle Scholar

Copyright information

© Birkhäuser-Verlag 2003

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WienWienAustria

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