Prevalent Dynamics at the First Bifurcation of Hénon-like Families

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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 languageEnglish
Pages (from-to)37-85
Number of pages49
JournalCommunications in Mathematical Physics
Volume312
Issue number1
DOIs
Publication statusPublished - 2012 May 1
Externally publishedYes

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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