Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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M. Astorg - F. Bianchi

Hyperbolicity and bifurcations in holomorphic families of polynomial skew products

created by bianchi on 12 May 2018
modified on 29 Mar 2024


Published Paper

Inserted: 12 may 2018
Last Updated: 29 mar 2024

Journal: American Journal of Mathematics
Volume: 145
Number: 3
Pages: 861-898
Year: 2023
Doi: 10.1353/ajm.2023.a897498

ArXiv: 1801.01460 PDF


We initiate a parametric study of families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)=(p(z), q(z,w))$ that extend to endomorphisms of $\mathbb{P}^2(\mathbb{C})$. Our aim is to study and give a precise characterization of the bifurcation current and the bifurcation locus of such a family. As an application, we precisely describe the geometry of the bifurcation current near infinity, and give a classification of the hyperbolic components. This is the first study of a bifurcation locus and current for an explicit and somehow general family in dimension larger than 1.

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