Conjugacy and counterexample in random iteration.
type: Item
instance of: scholarly article
title: Conjugacy and counterexample in random iteration.
zbMATH Open document ID: 1063.37041
author: Mark Comerford
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: This article is concerned with the structure of the Fatou set for random iteration of polynomials. ``Random iteration of polynomials'' is the composition of a sequence of polynomials which are not necessarily equal to each other, but which usually satisfy some uniform bounds. The Fatou and Julia sets for random iteration are defined similarly as for ordinary iteration. A fundamental theorem of Sullivan states that for every iterated rational map, every Fatou component is eventually periodic. As a consequence, every local limit function on the Fatou set must be either constant or univalent, and the diameters of the iterates of any Fatou component must be eventually periodic. In this article, counterexamples to both statements are given for random iteration of polynomials: there is a Fatou component on which there are both constant and nonconstant limit functions, and the diameters of the iterates of some Fatou component have positive lim sup but zero lim inf.
reviewed by: Dierk Schleicher
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37H99
zbMATH DE Number: 2101100
zbMATH Keywords: random iteration
zbMATH Keywords: Fatou set
zbMATH Keywords: polynomials
MaRDI profile type: Publication
full work available at URL: pjm.2003.211.69
OpenAlex ID: W1975119540
Wikidata QID: Q124968472
DOI: 10.2140/PJM.2003.211.69
description: scientific article; zbMATH DE number 2101100