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
-
(a)
fa* satisfies the Misiurewicz Condition,
-
(b)
fa* satisfies a backward Collet-Eckmann-like condition,
-
(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*.
Similar content being viewed by others
References
M. Benedicks and L. Carleson,On iterations of 1 −ax 2 on (−1, 1), Ann. Math.122 (1985), 1–25.
M. Benedicks and L. Carleson,The dynamics of the Hénon map, Ann. Math.133 (1991), 73–169.
P. Collet and J. P. Eckmann,Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston, 1980.
P. Collet and J. P. Eckmann,Positive Liapunov exponents and absolutely continuity, Ergodic Theory and Dynamical Systems3 (1983), 13–46.
A. Douady, private communication.
J. Guckenheimer,Sensitive dependence to initial conditions for one-dimensional maps, Commun. Math. Phys.70 (1979), 133–160.
M. Jakobson,Absolutely continuous measures for one-parameter families of one-dimensional maps, Commun. Math. Phys.81 (1981), 39–88.
M. V. Jakobson,Families of one-dimensional maps and nearby diffeomorphisms, Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986, pp. 1150–1160.
R. Mañé,Hyperbolicity, sinks and measure in one-dimensional dynamics, Commun. Math. Phys.112 (1987), 721–724.
M. Martens,The existence of σ-finite invariant measures, Applications to real 1-dimensional dynamics, preprint.
W. de Melo,Lectures on one-dimensional dynamics, IMPA (1989).
M. Misiurewicz,Absolutely continuous measures for certain maps of the interval, Publ. Math. IHES53 (1981), 17–51.
M. Martens, W. de Melo and S. van Strien,Julia-Fatou-Sullivan theory for real one-dimensional dynamics, preprint.
W. de Melo and S. van Strien,One-dimensional dynamics, preprint.
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.
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.
T. Nowicki and S. van Strien,Invariant measures exist under a summability condition for unimodal maps, preprint.
M. Rees,Positive measure sets of ergodic rational maps, Ann. Sci. Ec. Norm. Sup. 40 série19 (1986), 383–407.
M. Rychlik,Another proof of Jacobson’s theorem and related results, Ergodic Theory and Dynamical Systems8 (1988), 93–109.
D. Singer,Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math.35 (1978), 260.
S. van Strien,Hyperbolicity and invariant measures for general C2 interval maps satisfying the Misiurewicz condition, Commun. Math. Phys.128 (1990), 437–496.
M. Tsujii,Positive Lyapunov exponents in families of one-dimensional dynamical systems, preprint.
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.
Author information
Authors and Affiliations
Additional information
This work is partially supported by NSF.
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03008407