Prevalent dynamics at the first bifurcation of Henon-like families

TL;DR

This paper investigates the dynamics of Henon-like maps at the first bifurcation point, showing that almost all initial points diverge to infinity near this critical parameter, with a focus on non-recurrence of critical points.

Contribution

It establishes that the first bifurcation parameter is a full density point for divergence, using adapted parameter exclusion and volume control techniques from prior works.

Findings

The bifurcation parameter is a full Lebesgue density point.

Almost every initial point diverges to infinity at the bifurcation.

The critical points exhibit non-recurrence similar to Misiurewicz parameters.

Abstract

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.