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