Measure rigidity for some transcendental meromorphic functions
type: Item
instance of: scholarly article
title: Measure rigidity for some transcendental meromorphic functions
zbMATH Open document ID: 1252.37032
author: Agnieszka Badeńska
published in: Discrete and Continuous Dynamical Systems
publication date: 2012-07-04T00:00:00Z
reviewed by: Walter Bergweiler
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F15
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 6052999
zbMATH Keywords: meromorphic function
zbMATH Keywords: Julia set
zbMATH Keywords: conjugacy
zbMATH Keywords: rigidity
MaRDI profile type: Publication
full work available at URL: dcds.2012.32.2375
OpenAlex ID: W2314631611
DOI: 10.3934/DCDS.2012.32.2375
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review text: The author considers meromorphic functions \(f\) of the form \(f(z)=R(e^z)\) where \(R\) is a rational function such that \(\infty\) is an asymptotic value of \(f\). Let \(f\) and \(g\) be hyperbolic meromorphic functions of this form which are topologically conjugate on their Julia sets; that is, there exists a homeomorphism \(h:J(f)\to J(g)\) such that \(h\circ f=g\circ h\) on \(J(f)\backslash f^{-1}(\infty)\). Here \(J(\cdot)\) denotes the Julia set.NEWLINENEWLINEAssuming that \(f\) and \(g\) are non-exceptional in a certain sense, conditions are given that imply that \(h\) extends to an affine map conjugating \(f\) to \(g\). For example, it is shown that this is the case if \(h\) and \(h^{-1}\) are Lipschitz continuous or if \(h\) preserves the moduli of multipliers of periodic points; that is, if \(f^p(z)=z\), then \(|(f^p)'(z)|=|(g^p)'(h(z))|\).
description: scientific article; zbMATH DE number 6052999