Accessible fixed points on the boundary of stable domains
type: Item
instance of: scholarly article
title: Accessible fixed points on the boundary of stable domains
zbMATH Open document ID: 0894.30018
DOI: 10.1007/BF03322531
author: Walter Schmidt
published in: Results in Mathematics
publication date: 1998-09-13T00:00:00Z
review text: Let \(U\) be an invariant component of the Fatou set of a rational function \(R\) and \(z_0\in\partial U\) a weakly repelling fixed point of \(R\). We say that \(R\) is locally surjective at \(z_0\) with respect to \(U\) if, for any sufficiently small neighbourhood \(N\) of \(z_0\) we have \(R(N\cap U) =R(N) \cap R(U)\). This local surjectivity condition is certainly satisfied in the special case when \(U\) is completely invariant. The main result is that if \(R\) is locally surjective at the weakly repelling fixed point \(z_0 \in\partial U\) then \(z_0\) is an accessible boundary point. The result is due in this form to \textit{F. Prztycki} [Fundam. Math. 144, 259-278 (1994; Zbl 0812.58058)], but the present simpler proof refines a treatment of the case of simply-connected \(U\) given by \textit{L. Carleson} and \textit{T. W. Gamelin}, Complex dynamics (1993; Zbl 0782.30022).
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1112243
zbMATH Keywords: Fatou set
zbMATH Keywords: rational function
zbMATH Keywords: repelling fixed point
reviewed by: I. N. Baker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1112243