Recurrent Fuchsian groups whose Riemann surfaces have infinite dimensional spaces of bounded harmonic functions
type: Item
instance of: scholarly article
title: Recurrent Fuchsian groups whose Riemann surfaces have infinite dimensional spaces of bounded harmonic functions
zbMATH Open document ID: 0709.30038
author: John A. Velling
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: A Fuchsian group \(\Gamma\) acting on the unit disc D is called recurrent if for any positive measure subset \(A\subset S^ 1\) there exist infinitely many \(\gamma\in \Gamma\) such that \(A\cap \gamma A\) has positive Lebesgue measure. The paper under review contributes to the function theory corresponding to such groups, especially to the structure of the spaces of bounded harmonic functions on \({\mathcal R}=D/\Gamma\). Motivated by interesting examples, Taniguchi conjectured: If \(\Gamma\) is recurrent then \({\mathcal R}=D/\Gamma\) is in \(O^{\infty}_{HB}\). The author gives examples showing that this conjecture is false. - The surface \({\mathcal R}=D/\Gamma\) is said to be in \(O^ n_{HB}\) if up to sets of measure zero the action of \(\Gamma\) decomposes \(S^ 1\), into at most n disjoint positive measure ergodic components. The author constructs a surface \({\mathcal R}\in O^{\infty}_{HB}\setminus \cup^{\infty}_{n=1}O^ n_{HB}\) (see Theorem 2).
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 4165270
zbMATH Keywords: Fuchsian group
MaRDI profile type: Publication
full work available at URL: pjaa.65.211
OpenAlex ID: W2042147380
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DOI: 10.3792/PJAA.65.211
description: scientific article; zbMATH DE number 4165270