Local connectivity of the Julia set for geometrically finite rational maps
type: Item
instance of: scholarly article
title: Local connectivity of the Julia set for geometrically finite rational maps
zbMATH Open document ID: 0858.30021
author: Lei Tan
author: Yongcheng Yin
published in: Science in China. Series A
publication date: 1997-03-19T00:00:00Z
review text: The authors call a rational map \(f\) of \(\mathbb{C}\) of degree \(d>1\) geometrically finite if every critical point is either eventually periodic or attracted to an attracting or parabolic periodic cycle. The paper shows that if the Julia set \(J\) of such a map is connected, then \(J\) is also locally-connected. This was alaready known in the special cases when \(f\) is a polynomial or a hyperbolic rational map. In the case of a general geometrically finite rational map with non-connected \(J\) there remains a conjecture that every component of connectivity of \(J\) is locally connected. Combining the main theorem of the paper with a result of McMullen shows at least that every eventually periodic component of \(J\) is locally connected.
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 908288
zbMATH Keywords: local connectivity
zbMATH Keywords: hyperbolic
zbMATH Keywords: Julia set \(J\)
zbMATH Keywords: geometrically finite
reviewed by: I. N. Baker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 908288