Blenders near polynomial product maps of

  • Johan Taflin

    Université de Bourgogne Franche-Comté, Dijon, France
Blenders near polynomial product maps of $\mathbb{C}^2$ cover
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In this paper we show that if is a polynomial of degree possessing a neutral periodic point then a product map of the form can be approximated by polynomial skew products possessing special dynamical objects called blenders. Moreover, these objects can be chosen to be of two types: repelling or saddle. As a consequence, such a product map belongs to the closure of the interior of two different sets: the bifurcation locus of the space of holomorphic endomorphisms of degree of and the set of endomorphisms having an attracting set of non-empty interior. Similar techniques also give the first example of an attractor with non-empty interior or of a saddle hyperbolic set which is robustly contained in the small Julia set and whose unstable manifolds are all dense in . 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.

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Johan Taflin, Blenders near polynomial product maps of . J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3555–3589

DOI 10.4171/JEMS/1076