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

Ph Thieullen, C. Tresser, L. S. Young

Research output: Contribution to journalArticle

Abstract

Let f a a∈A be a C2 one-parameter family of non-flat unimodal maps of an interval into itself and a* 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 to x and a of f a n (x), respectively at the critical value and at a*, are comparable for large n. Then a* is a Lebesgue density point of the set of parameter values a 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*.

Original languageEnglish (US)
Pages (from-to)121-172
Number of pages52
JournalJournal d'Analyse Mathematique
Volume64
Issue number1
DOIs
StatePublished - Dec 1994

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Unimodal Map
Lyapunov Exponent
Critical value
Partial derivative
Henri Léon Lebésgue
Lebesgue Measure
Absolutely Continuous
Open set
Invariant Measure
Probability Measure
Interval
Family

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

Cite this

Positive Lyapunov exponent for generic one-parameter families of unimodal maps. / Thieullen, Ph; Tresser, C.; Young, L. S.

In: Journal d'Analyse Mathematique, Vol. 64, No. 1, 12.1994, p. 121-172.

Research output: Contribution to journalArticle

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