Fractional iteration and functional equations for functions analytic in the unit disk
type: Item
instance of: scholarly article
title: Fractional iteration and functional equations for functions analytic in the unit disk
zbMATH Open document ID: 1062.30026
DOI: 10.1007/BF03321854
author: Mark Elin
author: Simeon Reich
author: V. V. Goryainov
author: David Shoikhet
published in: Computational Methods and Function Theory
publication date: 2004-09-27T00:00:00Z
review text: Let \(\mathcal{P}\) denote the set of all holomorphic functions \(f\) in the open unit disk \(\mathbb D\) with the condition \(f(\mathbb D) \subset \mathbb D\). Then \(\mathcal{P}\) is a topological semigroup with respect to the operation of composition and the topology of locally uniform convergence. Denote by \(\mathcal{I}\) the set of all invertible elements of \(\mathcal{P}\), i.e. \(\mathcal{I}\) is the set of Möbius transformations which map the unit disk \(\mathbb D\) onto itself. A function \(f \in \mathcal{P}\) is called \textit{embeddable} if there exists a family \(\{f^t\}_{t \geq 0}\) in \(\mathcal{P}\) such that \(f^0=\text{id}\), \(f^1=f\), \(f^{t+s} = f^t \circ f^s\) for \(s\), \(t \geq 0\) and \(f^t \to \text{id}\) as \(t \to 0+\) locally uniformly in \(\mathbb D\). Therefore, the map \(t \mapsto f^t\) is a continuous homomorphism of the additive semigroup \(\mathbb R^+ = \{\, t \in \mathbb R : t \geq 0 \,\}\) into \(\mathcal{P}\), and thus \(\{f^t\}_{t \geq 0}\) is a one-parameter continuous semigroup in \(\mathcal{P}\). In this paper, the authors establish criteria for functions \(f \in \mathcal{P}\) to be embeddable. Here, the Abel and Schröder functional equations play an important role. To be more precise some further notation is necessary. Regarding iterates \(f^n\), \(n \in \mathbb N\) of \(f \in \mathcal{P}\) as a dynamical system, one has to consider the nature of fixed points of \(f\). In this connection an important role is played by the classical result of Denjoy and Wolff which asserts that for every \(f \in \mathcal{P} \setminus \mathcal{I}\) there exists a unique point \(q\), \(| q| \leq 1\) such that \(f^n \to q\) as \(n\to\infty\) locally uniformly in \(\mathbb D\). If \(q \in \mathbb D\), then \(f(q)=q\). In the case \(| q| =1\), there exist the angular limits \(f(q) := \lim\limits_{z \to q}{f(z)}\) and \(f'(q) := \lim\limits_{z \to q}{f'(z)}\) with \(f(q)=q\) and \(0 < f'(q) \leq 1\). This point \(q\) is called the \textit{Denjoy-W…
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 2103472
zbMATH Keywords: embeddability
zbMATH Keywords: fractional iteration
zbMATH Keywords: functional equation
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: Kœnigs function
zbMATH Keywords: semigroup of analyitc functions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03321854
OpenAlex ID: W2000339201
description: scientific article; zbMATH DE number 2103472