On the rate of convergence of parabolic semigroups of holomorphic functions
type: Item
instance of: scholarly article
title: On the rate of convergence of parabolic semigroups of holomorphic functions
zbMATH Open document ID: 1327.30026
author: Dimitrios Betsakos
published in: Analysis and Mathematical Physics
publication date: 2015-10-30T00:00:00Z
review text: Let \(\phi_t(z):\mathbb{D}\to \mathbb{D}\), \(t\geq 0\), be a one-parameter continuous semigroup of holomorphic functions. If for some \(t_0\geq 0\), the function \(f_{t_0}\) has no fixed point in \(\mathbb{D}\), then every function of the semigroup has no fixed point in \(\mathbb{D}\) and there exists a unique point \(\tau\in \partial \mathbb{D}\), called the Dejoy-Wolff point of the semigroup \(\phi_t\), such that for every \(z\in \mathbb{D}\), \(\lim_{t\to\infty}\phi_t(z)=\tau\). Without loss of generality, assume \(\tau=1\). The angular derivative \(\phi_t'(1)\) is either \(<1\) for every \(t>0\) or \(=1\) for every \(t>0\). In the former case, the semigroup is called hyperbolic, and in the latter case, it is called parabolic. Moreover, there exists a unique univalent function \(h:\mathbb{D}\to \mathbb{C}\) such that \(h(0)=0\) and \(\phi (t)=h^{-1}(h(z)+t)\). The domain \(\Omega =h(\mathbb{D})\) is called the associated planar domain of the semi-group. The type of the semigroup can be characterized by a geometric property of \(\Omega\). The semigroup is hyperbolic if and only if \(\Omega\) is contained in a horizontal strip. Let \(\nu (\Omega)\) denote the width of the horizontal strip. If the semigroup if parabolic, then \(\Omega\) may be contained in a horizontal half plane of the form \(\{z: \Im z>-\rho\}\) or \(\{z: \Im z<\rho\}\), \(\rho>0\). In a previous paper, the author estimated the rate of convergence if the semigroup is hyperbolic, in terms of the hyperbolic distance. In this paper, he estimates the rate of convergence of the semigroup for both cases in Euclidean distance, as follows. Let \((\phi_t)_{t\geq 0}\) be a semi-group of holomorphic self-maps of \(\mathbb{D}\) with Denjoy-Woff point \(1\) and associated planar domain \(\Omega\). Let \(z\in D\). (a) If the semigroup is hyperbolic, and \(\nu(\Omega)\) is the width of the smallest strip containing \(\Omega\), then there is a constant \(C_1>0\) such that, for every \(t>0\), \[ |\phi_t(z)-1|\leq…
reviewed by: Tao Chen
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F99
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 6502336
zbMATH Keywords: semigroup of holomorphic functions
zbMATH Keywords: Denjoy-Wolff point
zbMATH Keywords: harmonic measure
MaRDI profile type: Publication
full work available at URL: s13324-014-0095-8
OpenAlex ID: W2006280225
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DOI: 10.1007/S13324-014-0095-8
description: scientific article; zbMATH DE number 6502336