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

## TL;DR

This paper demonstrates that bifurcating polynomials in two complex variables can be approximated by special polynomial skew products with blenders, revealing new insights into complex dynamical systems and bifurcation phenomena.

## Contribution

It introduces the approximation of bifurcating polynomials by skew products with blenders of specific types, linking bifurcation loci and attracting sets in complex dynamics.

## Key findings

- Product maps can be approximated by skew products with blenders.
- Blenders can be chosen as repelling or saddle types.
- Bifurcation locus is in the closure of maps with attracting sets of interior.

## 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 H\'enon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.