Prevalent dynamics at the first bifurcation of Henon-like families

Details Prevalent dynamics at the first bifurcation of Henon-like families Prevalent dynamics at the first bifurcation of Henon-like families

2010-11-18

arxiv.org/abs/1011.4200v1

Mathematics

Dynamical Systems

45 pages, 6 figures

Scientific paper

We study the dynamics of strongly dissipative H\'enon-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 "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.

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