### 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) |
---|---|

Pages (from-to) | 121-172 |

Number of pages | 52 |

Journal | Journal d'Analyse Mathematique |

Volume | 64 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1994 |

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### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

### Cite this

*Journal d'Analyse Mathematique*,

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

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

Research output: Contribution to journal › Article

*Journal d'Analyse Mathematique*, vol. 64, no. 1, pp. 121-172. https://doi.org/10.1007/BF03008407

}

TY - JOUR

T1 - Positive Lyapunov exponent for generic one-parameter families of unimodal maps

AU - Thieullen, Ph

AU - Tresser, C.

AU - Young, L. S.

PY - 1994/12

Y1 - 1994/12

N2 - 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*.

AB - 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*.

UR - http://www.scopus.com/inward/record.url?scp=0001748121&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001748121&partnerID=8YFLogxK

U2 - 10.1007/BF03008407

DO - 10.1007/BF03008407

M3 - Article

VL - 64

SP - 121

EP - 172

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -