### 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 |
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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 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics