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Some properties of absolutely continuous invariant measures on an interval

Published online by Cambridge University Press:  19 September 2008

François Ledrappier
François Ledrappier*
Affiliation:
Laboratoire de calcul des probabilités, Université Paris, France
*
†Dr F. Ledrappier, Laboratoire de calcul des probabilités, Université Paris VI Tour 56, 4 Place Jussieu, 75230 Paris cedex 05, France.
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Abstract

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We are interested in ergodic properties of absolutely continuous invariant measures of positive entropy for a map of an interval. We prove a Bernoulli property and a characterization by some variational principle.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 1 , March 1981 , pp. 77 - 93
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Benettin, G., Casartelli, M., Galgani, L., Giorgilli, A. & Strelcyn, J.-M.. On the reliability of numerical studies of stochasticity. II. Identification of time averages. Il nuovo Cimento 50 B (2) (1979), 211–232.CrossRefGoogle Scholar
[2]Bowen, R.. A horseshoe with positive measure. Inventiones Math. 29 (1975), 203–204.CrossRefGoogle Scholar
[3]Bowen, R.. Bernoulli maps of the interval. Israel J. Maths 28 (1977), 161–168.CrossRefGoogle Scholar
[4]Bowen, R.. Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69 (1979), 1–18.CrossRefGoogle Scholar
[5]Bowen, R. & Ruelle, D.. The ergodic theory of axiom A flows. Inventiones Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[6]Bunimovic, L. A. & Sinai, Y. G.. Stochastic attractors and Lorenz model. In Nonlinear Waves (in Russian) Nauka Press, 1979.Google Scholar
[7]Fathi, A., Herman, M. & Yoccoz, J.-C.. Seminar notes, in preparation.Google Scholar
[8]Hofbauer, F.. Maximal measures for simple piecewise monotonic transformations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 52 (1980), 289–300.CrossRefGoogle Scholar
[9]Jacobs, K.. Lecture Notes on Ergodic Theory. Aarhus Universitët, 1963.Google Scholar
[10]Jakobson, M. V., Construction of invariant measures absolutely continuous with respect to dx for some maps of the interval. In Global Theory of Dynamical Systems. Springer Lecture Notes in Math. no. 819. Springer: Berli, 1980.Google Scholar
[11]Katok, A. & Strelcyn, J.-M.. Invariant manifold for smooth maps with singularities. Preprint.Google Scholar
[12]Ledrappier, F.. Principe variationnel et systèmes dynamiques symboliques. Z. Wahrscheinlichkeitstheorie verw. Gebiete 30 (1974), 185–202.CrossRefGoogle Scholar
[13]Ledrappier, F.. Sur la condition de Bernoulli faible et ses applications. Théorie ergodique; Rennes 73/74. Springer Lecture Notes in Math. no. 532. Springer: Berlin, 1976.Google Scholar
[14]Ledrappier, F. & Walters, P.. A relativised variational principle for continuous transformations. J. London Math. Soc. (2) 16 (1977), 568–576.CrossRefGoogle Scholar
[15]Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. preprint: IHES, 1979.Google Scholar
[16]Ornstein, D. S.. Ergodic Theory, Randomness and Dynamical Systems. Yale University Press: New Haven, 1974.Google Scholar
[17]Ornstein, D. S. & Weiss, B.. Geodesic flows are Bernoullian. Israel J. Math. 14 (1973), 184–198.CrossRefGoogle Scholar
[18]Pesin, Y. B.. Families of invariant manifolds corresponding to non-vanishing characteristic exponents. Izv. Akad. nauk. SSSR Ser. Mat. 40 (1976), 1332–1379.Google Scholar
Math USSR Izv. 10 (1976), 1261–1305.CrossRefGoogle Scholar
[19]Pesin, Y. B.. Description of the II-partition of a diffeomorphism with invariant smooth measure. Mat. Zametki 21 (6) (1977), 29–44.Google Scholar
[20]Ratner, M.. Anosov flows are also Bernoullian. Israeli. Math. 17 (1974), 380–391.CrossRefGoogle Scholar
[21]Rohlin, V. A.. On the fundamental ideas in measure theory. Mat. Sb. 25 (1949), 107–150.Google Scholar
Amer. Math. Soc. Trans. (1) 10 (1962), 1–54.Google Scholar
[22]Rohlin, V. A.. Exact endomorphism of a Lebesgue space. Izv. Akad. nauk. SSSR Ser. Mat. 25 (1961), 499–530.Google Scholar
Amer. Math. Soc. Trans. (2) 39 (1964), 1–36.Google Scholar
[23]Rohlin, V. A.. Lectures on the entropy theory of transformations with an invariant measure. Usp. Mat. nauk. 22 (5)(1967), 3–56.Google Scholar
Russian Math. Surveys 22 (5) (1967), 1–52.Google Scholar
[24]Ruelle, D.. A measure associated with axiom A attractors. Amer. I. Math. 98 (1975), 289–294.Google Scholar
[25]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978), 83–87.CrossRefGoogle Scholar
[26]Ruelle, D.. Sensitive dependence on initial conditions and turbulent behaviour of dynamical systems. Ann. NY. Acad. Sci. 316 (1978), 408–416.CrossRefGoogle Scholar
[27]Ruelle, D.. Ergodic theory of differentiable dynamical systems Publ. math. IHES 50 (1979), 27–58.CrossRefGoogle Scholar
[28]Ruelle, D.. Characteristic exponents and invariant manifolds in Hilbert space. preprint: IHES, 1980.Google Scholar
[29]Ruelle, D. & Shub, M.. Stable manifolds for maps. In Global Theory of Dynamical Systems. Springer Lecture Notes in Math. no. 819. Springer: Berli, 1980.Google Scholar
[30]Sinaī, Y. G.. Classical systems with countable Lebesgue spectrum II. Izv. Akad. nauk. SSSR Ser. Mat. 30 (1966), 15–68.Google Scholar
Amer. Math. Soc. Trans (2) 68 (1968), 34–88.Google Scholar
[31]Sinaī, Y. G.. Markov partitions and C-diffeomorphisms. Funk. Analiz i Pri. 2 (1968), 61–82.Google Scholar
Func. Ana. Appl. 2 (1968), 64–89.Google Scholar
[32]Sinaī, Y. G.. Gibbs measure in ergodic theory. Uspehi Mat. nauk 27 (4) (1972), 21–63.Google Scholar
Russian Math. Surveys 27 (4) (1972), 21–69.Google Scholar
[33]Szlenk, W.. Some dynamical properties of certain mappings of an interval, part I and II. Preprint: IHES, 1980.Google Scholar
[34]Walters, P.. Equilibrium states for β-transformations and related transformations. Math. Z. 159 (1978), 65–88.CrossRefGoogle Scholar