Blenders near polynomial product maps of ${\mathbb{C}}^2$

Abstract

In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto (p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types : repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets : the bifurcation locus of $H_d({\mathbb{P}}^2)$ and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of Hénon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.

arXiv commentv2: add a result using a heterodimentional cycle between blenders