Dynamics on weakly pseudoconvex domains
type: Item
instance of: scholarly article
title: Dynamics on weakly pseudoconvex domains
zbMATH Open document ID: 0843.58080
author: Fyuao Ren
author: Wenjun Zhang
published in: Chinese Annals of Mathematics. Series B
publication date: 1996-08-21T00:00:00Z
review text: In the given paper the iterative behaviour of holomorphic selfmappings \(f\in H(\Omega, \Omega)\) on suitable domains \(\Omega\in \mathbb{C}^2\) is studied if \((f^n )_{n\in \mathbb{N}}\) (the sequence of iterates) is normal on \(\Omega\). In \S 2 the authors give the following characterization on weakly pseudo-convex domains if \(f\) has some nonwandering point \(p\in \Omega\): Either \(p\) is an attracting fixpoint of \(f\). Or \(f\) is an isomorphism on some submanifold \(S\) of dimension 1 and some subsequence of \((f^n)\) converges to the identity on \(S\). Or \(f\) is an automorphism on \(\Omega\) and some subsequence of \((f^n)\) converges to the identity on \(\Omega\). This is in some sense a generalization of some result of \textit{P. Hriljac} [Nonlinear Anal. Theory Methods Appl. 19, No. 8, 717-730 (1992; Zbl 0771.32014)]. \S 3 handles the natural question what happens without the assumption on the existence of some nonwandering point. The following theorem of Denjoy-Wolff-type is shown under the additional assumption that the boundary of \(\Omega\) of real-analytic: Either \((f^n)\) converges to some point in \(\overline {\Omega}\) uniformly on compact subsets of \(\Omega\). Or there exists some uniquely defined holomorphic retraction \(R\) in the set of limit functions of \((f^n)\) such that for any limit function \(g\) there exists an automorphism \(T\) on the submanifold \(R(\Omega)\) with \(g= T\circ R\).
reviewed by: N.Terglane
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32H50
zbMATH DE Number: 832248
zbMATH Keywords: iteration
zbMATH Keywords: weakly pseudo-convex domains
zbMATH Keywords: nonwandering point
MaRDI profile type: Publication
Recommended article: Q4882236
Recommended article: Q4319816
Recommended article: Q4840641
Recommended article: Iterates of holomorphic self-maps on pseudoconvex domains of finite and infinite type in \(\mathbb C^n\)
Recommended article: Dynamics of holomorphic self-maps of regular domains and the injectivity theorem
Recommended article: The Wolff-Denjoy iteration property in complex Banach spaces
Recommended article: Random iteration of holomorphic self-maps over bounded domains in \(C^ N\)
Recommended article: Analytic dynamics in some pseudoconvex domains
Recommended article: The Denjoy-Wolff theorem in \(\mathbb C^n\)
Recommended article: Q3744479
description: scientific article; zbMATH DE number 832248