On the rate of convergence of semigroups of holomorphic functions at the Denjoy-Wolff point
type: Item
instance of: scholarly article
title: On the rate of convergence of semigroups of holomorphic functions at the Denjoy-Wolff point
zbMATH Open document ID: 1467.30007
author: Dimitrios Betsakos
author: Manuel D. Contreras
author: Santiago Díaz-Madrigal
published in: Revista Matemática Iberoamericana
publication date: 2021-03-07T00:00:00Z
review text: For a semigroup \((\phi_t)\) of holomorphic self-maps of the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) with Denjoy-Wolff point \(\tau\), the authors study the behavior of \(|\phi_t(z)-\tau|\) as \(t\to\infty\). For \(\phi:\mathbb D\to\mathbb D\) which is holomorphic, neither the identity, nor an elliptic automorphism, there exists a unique Denjoy-Wolff point \(\tau\) such that \(\phi(\tau)=\tau\). In the paper, \(|\tau|=1\) and hence \(\phi(\tau)\) is the angular limit of \(\phi\) at \(\tau\). In this case, the number \[\alpha_{\phi}(\tau):=\liminf_{z\to\tau}\frac{1-|\phi(z)|}{1-|z|},\;\;\;|\tau|=1,\] is the boundary dilatation coefficient of \(\phi\). A semigroup \((\phi_t)\), \(t\geq0\), is a continuous homomorphism \(t\mapsto\phi_t\) with respect to composition, endowed with the topology of uniform convergence on compacta, and \(\tau\) is the Denjoy-Wolff point for all \(\phi_t\). There exists \(\lambda\geq0\), the spectral value of \((\phi_t)\), such that \(\alpha_{\phi_t}(\tau)=e^{-\lambda t}\), \(t\geq0\). A semigroup \((\phi_t)\) is hyperbolic if \(\alpha_{\phi_t}\neq0\), and it is parabolic if \(\alpha_{\phi_t}=0\). Let \(\Omega\) be an open set in \(\mathbb C\) and let \(\Phi_t\) be a group of automorphisms of \(\Omega\). A Koenigs function \(h:\mathbb D\to h(\mathbb D)\subset\Omega\) is univalent, \(h\circ\phi_t=\Phi_t\circ h\) and \(\cup_{t\geq0}\Phi_t^{-1}(h(\mathbb D))=\Omega\). The rate of convergence of \(\phi_t(z)\) to \(\tau\) is given in the following proposition. Proposition 4.1. Let \((\phi_t)\) be a hyperbolic semigroup in \(\mathbb D\) with Denjoy-Wolff point \(\tau\), \(|\tau|=1\), and let \(\lambda\geq0\) be its spectral value. Then \[\lim_{t\to\infty}\frac{1}{t}\log(1-\overline{\tau}\phi_t(z))=-\lambda.\] The main result for hyperbolic semigroups in the paper shows that, in some cases, it is possible to get a tighter rate of convergence to \(\tau\). The authors characterize hyperbolic semigroups for which \[\lim_{t\to\infty}e^{\lambda …
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30C80
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 7318497
zbMATH Keywords: semigroups of holomorphic functions
zbMATH Keywords: Denjoy-Wolff point
zbMATH Keywords: harmonic measure
reviewed by: Dmitri V. Prokhorov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/rmi/1179
OpenAlex ID: W3005726971
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DOI: 10.4171/RMI/1179
description: scientific article; zbMATH DE number 7318497