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Positive Lyapunov exponent for generic one-parameter families of unimodal maps

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Abstract

Letf a a∈A be a C2 one-parameter family of non-flat unimodal maps of an interval into itself anda* a parameter value such that

  1. (a)

    fa* satisfies the Misiurewicz Condition,

  2. (b)

    fa* satisfies a backward Collet-Eckmann-like condition,

  3. (c)

    the partial derivatives with respect tox anda of f na (x), respectively at the critical value and ata*, are comparable for largen.

Thena* is a Lebesgue density point of the set of parameter valuesa such that the Lyapunov exponent of fa at the critical value is positive, and fa admits an invariant probability measure absolutely continuous with respect to the Lebesgue measure. We also show that given fa* satisfying (a) and (b), condition (c) is satisfied for an open dense set of one-parameter families passing through fa*.

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References

  1. M. Benedicks and L. Carleson,On iterations of 1 −ax 2 on (−1, 1), Ann. Math.122 (1985), 1–25.

    Article  MathSciNet  Google Scholar 

  2. M. Benedicks and L. Carleson,The dynamics of the Hénon map, Ann. Math.133 (1991), 73–169.

    Article  MathSciNet  Google Scholar 

  3. P. Collet and J. P. Eckmann,Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston, 1980.

    MATH  Google Scholar 

  4. P. Collet and J. P. Eckmann,Positive Liapunov exponents and absolutely continuity, Ergodic Theory and Dynamical Systems3 (1983), 13–46.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Douady, private communication.

  6. J. Guckenheimer,Sensitive dependence to initial conditions for one-dimensional maps, Commun. Math. Phys.70 (1979), 133–160.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Jakobson,Absolutely continuous measures for one-parameter families of one-dimensional maps, Commun. Math. Phys.81 (1981), 39–88.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. V. Jakobson,Families of one-dimensional maps and nearby diffeomorphisms, Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986, pp. 1150–1160.

    Google Scholar 

  9. R. Mañé,Hyperbolicity, sinks and measure in one-dimensional dynamics, Commun. Math. Phys.112 (1987), 721–724.

    Article  MATH  Google Scholar 

  10. M. Martens,The existence of σ-finite invariant measures, Applications to real 1-dimensional dynamics, preprint.

  11. W. de Melo,Lectures on one-dimensional dynamics, IMPA (1989).

  12. M. Misiurewicz,Absolutely continuous measures for certain maps of the interval, Publ. Math. IHES53 (1981), 17–51.

    MATH  MathSciNet  Google Scholar 

  13. M. Martens, W. de Melo and S. van Strien,Julia-Fatou-Sullivan theory for real one-dimensional dynamics, preprint.

  14. W. de Melo and S. van Strien,One-dimensional dynamics, preprint.

  15. T. Nowicki,A positive Liapounov exponent of the critical value of S-unimodal mapping implies uniform hyperbolicity, Ergodic Theory and Dynamical Systems8 (1988), 425–435.

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Nowicki and S. van Strien,Absolutely continuous invariant measures for C2 unimodal maps satisfying the Collet-Eckmann condition, Invent. Math.93 (1988), 619–635.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Nowicki and S. van Strien,Invariant measures exist under a summability condition for unimodal maps, preprint.

  18. M. Rees,Positive measure sets of ergodic rational maps, Ann. Sci. Ec. Norm. Sup. 40 série19 (1986), 383–407.

    MATH  MathSciNet  Google Scholar 

  19. M. Rychlik,Another proof of Jacobson’s theorem and related results, Ergodic Theory and Dynamical Systems8 (1988), 93–109.

    Article  MATH  MathSciNet  Google Scholar 

  20. D. Singer,Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math.35 (1978), 260.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. van Strien,Hyperbolicity and invariant measures for general C2 interval maps satisfying the Misiurewicz condition, Commun. Math. Phys.128 (1990), 437–496.

    Article  MATH  Google Scholar 

  22. M. Tsujii,Positive Lyapunov exponents in families of one-dimensional dynamical systems, preprint.

  23. Ph. Thieullen, Ch. Tresser and L. S. Young,Exposant de lyapunov dans des familles à un paramètre d’applications unimodales, C. R. Acad. Sci. Paris315 (1992), SérieI, 69–72.

    MATH  MathSciNet  Google Scholar 

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This work is partially supported by NSF.

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Thieullen, P., Tresser, C. & Young, L.S. Positive Lyapunov exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64, 121–172 (1994). https://doi.org/10.1007/BF03008407

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  • DOI: https://doi.org/10.1007/BF03008407

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