Iteration theory in hyperbolic domains
type: Item
instance of: scholarly article
title: Iteration theory in hyperbolic domains
zbMATH Open document ID: 0906.30023
DOI: 10.1007/BF02819445
author: Fabio Vlacci
published in: Journal d'Analyse Mathématique
publication date: 1999-02-22T00:00:00Z
review text: Some of the deep results proved in 1981, 1984 by \textit{C. Cowen} [Trans. Am. Math. Soc. 265, 69-95 (1981; Zbl 0476.30017); ibid. 283, 685-695 (1984; Zbl 0542.30030)] about iteration and fixed-points of holomorphic self-maps of the disk \( \Delta\) are extended to hyperbolic domains of regular type. These are noncompact domains \(D\) lying on a compact Riemann surface \(\widehat X\) such that (i) every connected component of \(\partial D\) is either a Jordan curve or a single point and (ii) every such component can be separated from all the rest by an open subset of \(\widehat X\). Effectively this class of domains contains all finitely multiply connected hyperbolic domains. Naturally heavy use is made of the universal covering of \(D\) by \(\Delta\). A 1970 result of \textit{W. Pranger} [Aequationes Math. 4, 201-204 (1970; Zbl 0197.35004] is also generalized to \(D\). This is a characterization of sets \(\{g'(z_0): g\in \text{Hol} (\Delta)\), \(g\circ f= f\circ g\}\) arising from some locally univalent \(f\in\text{Hol} (\Delta, \Delta)\) which fixes \(z_0\). A representative Cowen result that is generalized concerns the existence of a fundamental set for each fixed-point-free \(f\in\text{Hol} (\Delta, \Delta)\). This is an open, connected, simply-connected \(V\subset \Delta\) which \(f\) maps injectively into itself and which devours the tails of \(f\)-orbits of all compact \(K\subset \Delta\): for all sufficiently large \(n\), the \(n\)th iterate of \(f\) maps \(K\) into \(V\).
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1187468
zbMATH Keywords: finite-connectivity
zbMATH Keywords: iteration
zbMATH Keywords: fixed-points
zbMATH Keywords: hyperbolic domains
reviewed by: Robert Burckel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1187468