6533b82afe1ef96bd128b797

RESEARCH PRODUCT

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

Johan Taflin

subject

PolynomialEndomorphismMathematics::Dynamical SystemsMathematics - Complex VariablesApplied MathematicsGeneral Mathematics010102 general mathematicsClosure (topology)BlendersattractorsDynamical Systems (math.DS)01 natural sciencesSet (abstract data type)CombinatoricsBifurcation locusProduct (mathematics)AttractorFOS: MathematicsComplex Variables (math.CV)0101 mathematics[MATH]Mathematics [math]Mathematics - Dynamical SystemsbifurcationsSaddleMathematics

description

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.

yearjournalcountryeditionlanguage
2021-04-15
http://arxiv.org/abs/1702.02115