### Abstract

We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter a* at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that a* is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that a* corresponds to the "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusion argument, we construct a set of "good parameters" having a* as a full density point. Adapting Benedicks & Viana's volume control argument on the other, we analyze Lebesgue typical dynamics corresponding to these good parameters.

Original language | English |
---|---|

Pages (from-to) | 37-85 |

Number of pages | 49 |

Journal | Communications in Mathematical Physics |

Volume | 312 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 May |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Prevalent Dynamics at the First Bifurcation of Hénon-like Families.** / Takahasi, Hiroki.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 312, no. 1, pp. 37-85. https://doi.org/10.1007/s00220-012-1442-y

}

TY - JOUR

T1 - Prevalent Dynamics at the First Bifurcation of Hénon-like Families

AU - Takahasi, Hiroki

PY - 2012/5

Y1 - 2012/5

N2 - We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter a* at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that a* is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that a* corresponds to the "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusion argument, we construct a set of "good parameters" having a* as a full density point. Adapting Benedicks & Viana's volume control argument on the other, we analyze Lebesgue typical dynamics corresponding to these good parameters.

AB - We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter a* at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that a* is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that a* corresponds to the "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusion argument, we construct a set of "good parameters" having a* as a full density point. Adapting Benedicks & Viana's volume control argument on the other, we analyze Lebesgue typical dynamics corresponding to these good parameters.

UR - http://www.scopus.com/inward/record.url?scp=84860354991&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860354991&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1442-y

DO - 10.1007/s00220-012-1442-y

M3 - Article

AN - SCOPUS:84860354991

VL - 312

SP - 37

EP - 85

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -