Nonwandering, nonrecurrent Fatou components in \({\mathbb{P}}^2\).
type: Item
instance of: scholarly article
title: Nonwandering, nonrecurrent Fatou components in \({\mathbb{P}}^2\).
zbMATH Open document ID: 1058.37036
author: Brendan J. Weickert
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: Let \(f\) be a holomorphic endomorphism of the projective plane \(\mathbb{P}^2\) and \(\Omega\) be a fixed Fatou component. We assume that \(\Omega\) is nonrecurrent, i.e., the orbit of its points go to the boundary of \(\Omega\). Let \(h\) be a limit of the sequence of the iterates of \(f\) on \(\Omega\) and note \(\Sigma\) the set \(h(\Omega)\). The author shows that \(\Sigma\) is either a fixed-point or a hyperbolic Riemann surface. In the latter case, the restriction of \(f\) on \(\Sigma\) can be lifted to an holomorphic map \(g\) on the unit disc \(D\), and there are two possibilities. Either \(g\) is an irrational rotation of \(D\), or the sequence of iterates \(g^n\) converges to the boundary of \(D\). The author also proves the following independant result : the basins of attraction of holomorphic endomorphisms of \(\mathbb{P}^k\) are taut, i.e., the family of holomorphic discs (into such a basin) is normal.
reviewed by: Christophe Dupont
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 2101117
zbMATH Keywords: holomorphic dynamics
zbMATH Keywords: Fatou component
MaRDI profile type: Publication
full work available at URL: pjm.2003.211.391
OpenAlex ID: W1992535961
DOI: 10.2140/PJM.2003.211.391
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description: scientific article; zbMATH DE number 2101117