Absolutely continuous conjugacies of Blaschke products. III
type: Item
instance of: scholarly article
title: Absolutely continuous conjugacies of Blaschke products. III
zbMATH Open document ID: 0809.30028
DOI: 10.1007/BF03008429
author: David . H. Hamilton
published in: Journal d'Analyse Mathématique
publication date: 1994-05-18T00:00:00Z
review text: Let \(f\) be a Blaschke product in the unit disk \(\mathbb{D}\). Its Julia set is the entire unit circle \(S\) if and only if it is ergodic. Let \(\varphi\) denote an absolutely continuous homeomorphism of \(S\) onto itself. In an earlier paper the author showed that \(\varphi \circ f \circ \varphi^{- 1}\) can be a Blaschke product only in the trivial case that \(\varphi\) is a Möbius transformation. The present paper studies the case that \(f\) is not ergodic. It is shown that there always exists \(\varphi\) such that \(f^* = \varphi \circ f \circ \varphi^{-1}\) is a finite Blaschke product but \(f^* \neq Q \circ f \circ Q^{-1}\) on \(S\) for all Möbius transformations \(\varphi\) of \(\mathbb{D}\) onto itself. The same result holds if ``finite Blaschke product'' is replaced by ``inner functions''. The proof uses the fact that there is an inner function \(g\) fixing 0 and a Möbius transformation \(T\) such that \(g \circ f = T \circ g\).
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 569426
zbMATH Keywords: Blaschke product
zbMATH Keywords: Julia set
zbMATH Keywords: ergodic
zbMATH Keywords: absolutely continuous
zbMATH Keywords: Möbius transformation
zbMATH Keywords: inner function
reviewed by: Christian Pommerenke
MaRDI profile type: Publication
cites work: Q4157390
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description: scientific article; zbMATH DE number 569426