Preprint D79/2010
Prevalent dynamics at the first bifurcation of Henon-like families

Hiroki Takahasi

**Keywords: **

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 ``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.