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Positive Lyapunov exponents in families of one dimensional dynamical systems

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We give a generalization of Jakobson's theorem [Ja] and a precise estimate for the density of parameters corresponding to systems with absolutely continuous invariant measures.

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References

  • [BC1] Benedicks, M., Carleson, L.: On Iterations of 1−ax 2 on (−1, 1). Ann. Math., II. Ser.122, 1–24 (1985)

    Google Scholar 

  • [BC2] Benedicks, M., Carleson, L.: The Dynamics of the Hénon Map. Ann. Math., II. Ser.133, 73–169 (1991)

    Google Scholar 

  • [BY] Benedicks, M., Young, L.-S.: Absolutely Continous Invariant Measures and Random Perturbations for Certain One Dimensional Maps. Ergodic Theory Dyn. Syst. (to appear)

  • [BL] Blokh, A.M., Lyubich, M.Yu.: Non-existence of Wandering Interval and Structure of Topological Attractors of One Dimensional Dynamical Systems II. The Smooth Case. Ergodic Theory Dyn. Syst.,9, 751–758 (1989)

    Google Scholar 

  • [Ja] Jakobson, M.V.: Absolutely Continuous Invariant Measures for One Parameter Families of One Dimensional Maps. Commun. Math. Phys.81, 39–88 (1981)

    Google Scholar 

  • [Jo] Johnson, S.: Continuous Measures in One Dimension. Commun. Math. Phys.122, 293–320 (1989)

    Google Scholar 

  • [Le] Ledrappier, F.: Some Properties of Absolutely Continuous Invariant Measures on an Interval. Ergodic Theory Dyn. Sys.1, 77–93 (1981)

    Google Scholar 

  • [Ma] Mañé, R.: Hyperbolicity, Sinks and Measure in One Dimensional Dynamics. Commun. Math. Phys.100, 495–524 (1985)

    Google Scholar 

  • [MMS] Martens, M., deMelo, W., van Strien, S.: Julia-Fatou-Sullivan Theory for One Dimensional Dynamics. Acta Math.168, 273–318 (1992)

    Google Scholar 

  • [MV] Mora, L., Viana, M.: Abundance of Strange Attractors. Acta. Math. (to appear)

  • [NS] Nowicki, T., van Strien, S.: Absolutely Continuous Measures for Collet-Eckmann Maps without Schwarzian Derivative Conditions. Invent. Math.93, 619–635 (1988)

    Google Scholar 

  • [R] Rychlik, M.: Another Proof of Jakobson's Theorem and Related Results. Ergodic Theory Dyn. Syst.8, 93–110 (1986)

    Google Scholar 

  • [T1] Tsujii, M.: Weak Regularity of Lyapunov Exponent in One Dimensional Dynamics. (Preprint)

  • [T2] Tsujii, M.: A Proof of Benedicks-Carleson-Jakobson Theorem. (Preprint)

  • [T3] Tsujii, M.: Small Random Perturbations of One Dimensional Dynamical Systems and Margulis-Pesin entropy Formula. Random Comput. Dyn.1, 59–89 (1992)

    Google Scholar 

  • [Y] Young, L.-S.: Decay of Correlations for Certain Quadratic Maps. (Preprint)

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Author notes

  1. Masato Tsujii

    Present address: Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro-ku, 152, Tokyo, Japan

Authors and Affiliations

  1. Department of Mathematics, Kyoto University, 606, Kyoto, Japan

    Masato Tsujii

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  1. Masato Tsujii
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Dedicated to the memory of Prof. Gikō Ikegami

Oblatum 11-III-1992

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Tsujii, M. Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent Math 111, 113–137 (1993). https://doi.org/10.1007/BF01231282

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