Dynamics of transcendental entire maps on Berkovich affine line
type: Item
instance of: scholarly article
title: Dynamics of transcendental entire maps on Berkovich affine line
zbMATH Open document ID: 1283.37077
author: Shilei Fan
author: Yuefei Wang
published in: Journal of Dynamics and Differential Equations
publication date: 2013-05-06T00:00:00Z
review text: In this article, the authors study the dynamics of transcendental entire maps on the Berkovich affine line \(\mathbb{A}_{\mathrm{Berk}}^1\) for an algebraically closed field \(K\) which is complete with respect to a non-trivial absolute non-Archimedean value \(|\cdot|\). For a transcendental entire map \(f: \mathbb{A}_{\mathrm{Berk}}^1 \to\mathbb{A}_{\mathrm{Berk}}^1\), the authors show that the Julia set \({\mathcal{J}}_f\) is the closure of the set of repelling periodic points of \(f\) in \(\mathbb{A}_{\mathrm{Berk}}^1\). Further, for a subset \(U\) of the Fatou set \({\mathcal{F}}_f\), the complement of \({\mathcal{J}}_f\), which is multiply connected, they also show that \(U\) is a wandering component, its image under the \(n\)th iterate of \(f\), \(f^n(U)\), is multiply connected, and for every \(x\in U\), \(\lim_{n\to \infty} f^n(x)\) diverges; moreover, if \(U\) is a finitely multiply connected component, it is also proved that there exists a positive integer \(n\) large enough such that \(f^n(U)\) is \(2\)-connected.
reviewed by: Shigeki Matsutani
Mathematics Subject Classification ID: 37P50
Mathematics Subject Classification ID: 37P40
Mathematics Subject Classification ID: 11S82
Mathematics Subject Classification ID: 30D99
zbMATH DE Number: 6160883
zbMATH Keywords: transcendental entire map
zbMATH Keywords: non-Archimedean field
zbMATH Keywords: Berkovich affine line
zbMATH Keywords: Fatou set
zbMATH Keywords: Julia set
zbMATH Keywords: dynamical system
MaRDI profile type: Publication
full work available at URL: s10884-012-9281-2
OpenAlex ID: W2087214600
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DOI: 10.1007/S10884-012-9281-2
description: scientific article; zbMATH DE number 6160883