Positive Lyapunov exponent for generic one-parameter families of unimodal maps |
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Authors: | Ph. Thieullen C. Tresser L. S. Young |
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Affiliation: | 1. Département de Mathématiques, Université Paris Sud, 91405, Orsay Cedex, France 2. Department of Mathematics, University of Arizona, 85721, Tucson, AZ, USA 3. I.B.M., P.O. Box 210, 10958, Yorktown Heights, NY, USA 4. Department of Mathematics, U.C.L.A., 90024, Los Angeles, CA, USA
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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 - fa* satisfies the Misiurewicz Condition,
- fa* satisfies a backward Collet-Eckmann-like condition,
- the partial derivatives with respect tox anda of f a n (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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