Measures Invariant under Mappings of the Unit Interval

  • Pierre Collet
  • Jean-Pierre Eckmann


It has been recognized in the last few years that dynamical systems with few degrees of freedom can play an important role in the description of some physical systems which behave in a chaotic way. The rather simple one dimensional dynamical systems can also be used to test some ideas about higher dimensional systems. The question of the statistical description of chaotic motions was already raised several decades ago ([U]) and after the discovery of the ergodic theorems it became obvious that an important notion is that of invariant measure. For a given dynamical system, there are in general many invariant measures, and one is led to the problem of choosing the relevant one (if any).


Invariant Measure Central Limit Theorem Parameter Family Ergodic Theorem Ergodic Property 
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  1. A.
    R.F. Adler, F -expansions revisited, in “Recent Advances in Topological Dynamics”,Lecture Notes in Mathematics, 318 Springer -Verlag, Berlin, Heidelberg, New York 1973.Google Scholar
  2. An.
    D.V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of mathematics, 90 (1967)Google Scholar
  3. B1.
    R. Bowen, Equilibrium states and the ergodic theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, Springer Verlag, Berlin, Heidelberg, New York, 1975.Google Scholar
  4. B2.
    R. Bowen, Invariant measures for Markov maps of the interval Commun. Math. Phys., 69, (1979) 1–17.MathSciNetADSCrossRefMATHGoogle Scholar
  5. B3.
    R. Bowen, Bernouilli maps of the interval. Israël J. Math. 28 (1977) 161–168.MathSciNetADSMATHGoogle Scholar
  6. B.C.
    M. Benedicks, L. Carleson, On iterations of 1 - ax2 on (-1,1), Preprint, Institut Mittag Leffler 1983.Google Scholar
  7. CE1.
    P. Collet, J-P. Eckmann,“Iterated maps on the interval as dynamical systems”, Birkhaüser, Basel, Boston, 1980.Google Scholar
  8. CE2.
    P. Collet, J-P. Eckmann, Positive Lyapunov exponents and absolute continuity for maps of the interval. Ergodic theory and dynamical systems, to appear.Google Scholar
  9. CE3.
    P. Collet, J-P. Eckmann, On the abundance of aperiodic behaviour for maps on the interval. Commun. Math. Phys., 73, (1980) 115–160.MathSciNetADSCrossRefMATHGoogle Scholar
  10. CEL.
    P. Collet, J-P. Eckmann, 0.E. Lanford III: Universal properties of maps on an interval. Commun. Math. Phys., 76, (1980) 211–254.MathSciNetADSCrossRefMATHGoogle Scholar
  11. C.F.
    P. Collet, P. Ferrero, In preparation.Google Scholar
  12. C.L.
    P. Collet, Y. Lévy, Ergodic properties of the Lozi Mappings. Preprint Ecole Polytechnique, Paris 1983.Google Scholar
  13. C.T.
    P. Collet, C. Tresser, In preparation.Google Scholar
  14. Fe.
    W.F. Feller,“An introduction to probability theory and applications”,John Wiley and Sons 1971.Google Scholar
  15. G.
    J. Guckenheimer, Renormalization of one dimensional mappings and strange attractors. Unpublished.Google Scholar
  16. H.K.
    F. Hofbauer, G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Zeit. 180, (1982) 119–140.MathSciNetCrossRefMATHGoogle Scholar
  17. J.
    M. Jakobson, Absolutely continuous invariant measures for one parameter families of one dimensional maps. Commun. Math. Phys. 81, (1981) 39–88.MathSciNetADSCrossRefMATHGoogle Scholar
  18. K.
    G. Keller, Stochastic stability in some chaotic dynamical systems. Preprint, Univ. Heidelberg 1981.Google Scholar
  19. L.
    O.E. Lanford III, CIME Lecture 1978.Google Scholar
  20. La.Y1
    A. Lasota, J. Yorke, On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973) 481–488.MathSciNetCrossRefGoogle Scholar
  21. La.Y2
    A. Lasota, J. Yorke, Exact dynamical systems and the Frobenius-Perron operator. Trans. Amer. Math. Soc., 273 (1982) 375–384.MathSciNetCrossRefMATHGoogle Scholar
  22. Le.
    F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. and Dynam. Sys. 1, (1981) 77–93.MathSciNetCrossRefMATHGoogle Scholar
  23. Li.Y.T.Y.
    Li, J. Yorke, Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978) 183–193.MathSciNetCrossRefMATHGoogle Scholar
  24. M.
    P. Manneville, Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems. Journ. de Phys. Paris 41, (1980) 1235–1243.MathSciNetGoogle Scholar
  25. Me.
    C. Meunier, Continuity of type I Intermittency from a measure theoretical point of view. Preprint, Ecole Polytechnique, Palaiseau 1982.Google Scholar
  26. Mi.
    M. Misiurewicz, Absolutely continuous measure for certain maps of an interval. Publ. Sci. IRES. 53 (1981) 17–52.MathSciNetMATHGoogle Scholar
  27. M.T.
    J. Milnor, P. Thurston, On iterated maps of the interval, I, II. Preprint, Princeton University 1977.Google Scholar
  28. N.
    Z. Nitecki, Topological Dynamics on the interval,“Ergodic theory and dynamical systems”,Vol. II, A. Katok ed. Birkhäuser, Boston, Basel 1981.Google Scholar
  29. P.
    C. Preston, Iterates of maps on an interval, Lecture Notes in Mathematics 999. Springer-Verlag, Berlin, Heidelberg, New York 1983.Google Scholar
  30. Pi.
    G. Pianigiani, First return map and invariant measures, Israël J. Math., 35 (1980) 32–48.MathSciNetCrossRefMATHGoogle Scholar
  31. Re.
    A. Renyi, Representations of real numbers and their ergodic properties. Acta. Math. Akad. Sci. Hungar., 8 (1957) 477–493.MathSciNetCrossRefMATHGoogle Scholar
  32. R1.
    D. Ruelle,“Thermodynamic formalism”,Addison-Wesley, London, Amsterdam 1978.Google Scholar
  33. R2.
    D. Ruelle, Applications conservant une mesure absolument continue par rapport A dx sur [0,1] Commun. Math. Phys. 55 (1977) 47–51.MathSciNetADSCrossRefMATHGoogle Scholar
  34. R.
    J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. The Ann. of Prob. 11, (1983) 772–788.MathSciNetCrossRefMATHGoogle Scholar
  35. Ry.
    M. Rychlik, Invariant measures for piecewise monotonic, piecewise C1+ transformations. To appear.Google Scholar
  36. S.
    F. Schweiger, Number theoretical endomorphisms with a finite invariant measure. Israël J. Math. 21 (1975) 308–318.Google Scholar
  37. Sz.
    W. Szlenk, Some dynamical properties of certain differenttiable mappings of an interval I.II Preprint IHES 1980.Google Scholar
  38. T.
    M. Thaler, Tranformation on [0,1] with infinite invariant measures. Preprint, University of Salzburg 1983. Estimates of the invariant densities of endomorphisms with indifferent fixed points. Israel J. Math., 37 (1980) 303–314.MathSciNetGoogle Scholar
  39. U.
    S. Ulam,’A collection of mathematical problems“, Interscience New York 1960.Google Scholar
  40. Wo1.
    S. Wong, Hölder continuous derivatives and ergodic theory. Proc. London Math. Soc. 22 (1980) 506–520.MathSciNetCrossRefMATHGoogle Scholar
  41. Wo2.
    S. Wong, A central limit theorem for piecewise monotonic mappings of the unit interval. Ann. of Prob. 7 (1979) 500–514.CrossRefMATHGoogle Scholar
  42. W1.
    P. Walters, Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math Soc. 236 (1978) 121–153.MathSciNetCrossRefGoogle Scholar
  43. W2.
    P. Walters, Ergodic theory introductory lectures. Lecture Notes in Mathematics 458, Springer-Verlag, Berlin, Heidelberg, New York 1975.Google Scholar
  44. Y.
    K. Yosida, “Functional Analysis”,Springer-Verlag Berlin, Heidelberg, New York 1968.Google Scholar

Copyright information

© Plenum Press, New York 1985

Authors and Affiliations

  • Pierre Collet
    • 1
  • Jean-Pierre Eckmann
    • 2
  1. 1.Centre de Physique ThéoriqueEcole PolytechniquePalaiseau CedexFrance
  2. 2.Dept. de Physique ThéoriqueGenève 4Suisse

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