JournalsjemsVol. 23, No. 11pp. 3555–3589

Blenders near polynomial product maps of C2\mathbb{C}^2

  • 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 pp is a polynomial of degree d2d\geq2 possessing a neutral periodic point then a product map of the form (z,w)(p(z),q(w))(z,w)\mapsto(p(z),q(w)) can be approximated by polynomial skew products (z,w)(p~(z,w),q(w))(z,w)\mapsto(\tilde{p}(z,w),q(w)) 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 dd of P2\mathbb{P}^2 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 P2\mathbb{P}^2. 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.

Cite this article

Johan Taflin, Blenders near polynomial product maps of C2\mathbb{C}^2. J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3555–3589

DOI 10.4171/JEMS/1076