Composite Julia sets generated by infinite polynomial arrays.
type: Item
instance of: scholarly article
title: Composite Julia sets generated by infinite polynomial arrays.
zbMATH Open document ID: 1040.37033
author: M. Klimek
author: Marta Kosek
published in: Bulletin des Sciences Mathématiques
publication date: 2004-01-26T00:00:00Z
review text: Let \({\mathcal F}\) be a family of proper polynomial mappings \(P: \mathbb{C}^N\to \mathbb{C}^N\) with Lojasiewicz exponent \(> 1\). Then it is known, the set function \[ {\mathcal H}_{\mathcal F}:{\mathcal R}\ni K\mapsto \Biggl(\bigcup_{P\in{\mathcal F}} P^{-1}(K)\Biggr)^\Lambda\in{\mathcal R}\tag{1} \] becomes a contraction of \({\mathcal R}\) and hence has a unique fixed point. Here, \({\mathcal R}\) denotes the family of all polynomial convex pluriregular compact subsets of \(\mathbb{C}^N\) and the exponent \(\Lambda\) denotes the operator taking the polynomially convex hull of the set enclosed in the big parentheses. In this paper, the authors show that under suitable assumptions the mapping (1) remains a contraction of \({\mathcal R}\) even for some class of \({\mathcal F}\) containing infinitely many polynomial mappings. Moreover, they show that mappings of this type preserve the Hölder continuity property of compact sets.
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 2032711
zbMATH Keywords: Julia sets
zbMATH Keywords: pluricomplex Green function
zbMATH Keywords: analytic multifunctions
Wikidata QID: Q121600056
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: j.bulsci.2003.08.002
OpenAlex ID: W2010037006
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DOI: 10.1016/J.BULSCI.2003.08.002
description: scientific article; zbMATH DE number 2032711