### Abstract

Let f_{ a a∈A} be a C^{2} one-parameter family of non-flat unimodal maps of an interval into itself and a* a parameter value such that (a) f_{a*} satisfies the Misiurewicz Condition, (b) f_{a*} 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 f_{a} at the critical value is positive, and f_{a} admits an invariant probability measure absolutely continuous with respect to the Lebesgue measure. We also show that given f_{a*} satisfying (a) and (b), condition (c) is satisfied for an open dense set of one-parameter families passing through f_{a*}.

Original language | English (US) |
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Pages (from-to) | 121-172 |

Number of pages | 52 |

Journal | Journal d'Analyse Mathématique |

Volume | 64 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1994 |

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

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## Cite this

*Journal d'Analyse Mathématique*,

*64*(1), 121-172. https://doi.org/10.1007/BF03008407