Dynamical rigidity of transcendental meromorphic functions
type: Item
instance of: scholarly article
title: Dynamical rigidity of transcendental meromorphic functions
zbMATH Open document ID: 1277.37077
DOI: 10.1088/0951-7715/25/8/2337
author: Bartłomiej Skorulski
author: Mariusz Urbański
published in: Nonlinearity
publication date: 2012-08-24T00:00:00Z
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F15
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37F35
Mathematics Subject Classification ID: 37F30
zbMATH DE Number: 6071986
zbMATH Keywords: iteration
zbMATH Keywords: tame meromorphic function
zbMATH Keywords: Julia set
zbMATH Keywords: conjugacy
zbMATH Keywords: bi-Lipschitz
zbMATH Keywords: dynamical rigidity
zbMATH Keywords: nice sets
zbMATH Keywords: iterated function system
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1088/0951-7715/25/8/2337
OpenAlex ID: W2005893017
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review text: This interesting paper is devoted to the study of the dynamics of iterates for transcendental meromorphic functions in the complex plane~\(\mathbb C\). The Fatou set \({\mathcal F}(f)\) of a meromorphic function \(f:{\mathbb C}\to\hat{\mathbb C}\) is the maximal open set on which the iterates \(f^{\circ n}\), \(n\in\mathbb N\), are all well-defined and form a normal family. The Julia set \({\mathcal J}(f):=\hat{\mathbb C}\setminus{\mathcal F}(f)\) is defined as its spherical complement. A meromorphic function is said to be tame, if its postcritical set NEWLINE\[NEWLINEPS(f):=\overline{\bigcup_{n=1}^{+\infty}f^{\circ n}(\text{sing} f^{-1})},NEWLINE\]NEWLINE where \(\text{sing} f^{-1}\) stands for the set of all singularities of~\(f^{-1}\), does not contain \({\mathcal J}(f)\). The main result of the paper, Theorem~5.1, is as follows.NEWLINENEWLINEIf two tame transcendental meromorphic functions \(f,\,g: {\mathbb C}\to\hat{\mathbb C}\) are conjugate on their Julia sets by a bi-Lipschitz homeomorphism NEWLINE\[NEWLINE{H:{\mathcal J}(f)\to{\mathcal J}(g)},\quad {\text{ i.e. }} H(f(z))=g(H(z)) \tag{*} NEWLINE\]NEWLINE for all~\(z\in{\mathcal J}(f)\setminus\{\infty\}\), then \(H\) extends to an affine map \(H:{\mathbb C}\to{\mathbb C}\); \(z\mapsto az+b\), for which~ (*) holds for all~\(z\in \mathbb C\).NEWLINENEWLINEThe methods used in the proof are interesting on their own. Continuing the study made in~[\textit{D.\ Mauldin, F. Przytycki} and the second author, Compos. Math. 129, No. 3, 273--299 (2001; Zbl 1106.37306)], the authors consider conformal iterated function systems (for the definition see~[\textit{D. Mauldin} and the second author, Proc. Lond. Math. Soc., III. Ser. 73, No. 1, 105--154 (1996; Zbl 0852.28005); Graph directed Markov systems. Geometry and dynamics of limit sets. Cambridge: Cambridge University Press (2003; Zbl 1033.37025)]) associated to the so-called nice sets of a tame meromorphic function, see [\textit{J. Rivera-Letelier}, Ergodic Theory Dyn. S…
reviewed by: Pavel Gumenyuk
description: scientific article; zbMATH DE number 6071986