The iteration of cubic polynomials. II: Patterns and parapatterns
type: Item
instance of: scholarly article
title: The iteration of cubic polynomials. II: Patterns and parapatterns
zbMATH Open document ID: 0812.30008
DOI: 10.1007/BF02392761
author: Bodil Branner
author: John Hamal Hubbard
published in: Acta Mathematica
publication date: 1995-05-16T00:00:00Z
review text: For Part I see the authors in ibid. 160, No. 3/4, 143--206 (1988; Zbl 0668.30008). This extensive paper (rather the monograph) consists of 12 chapters. In 1918 and 1920 G. Julia and P. Fatou proved that the Julia set for a polynomial \(P\) is connected if and only if none of the critical points escape to infinity under iteration and the Julia set for a polynomial \(P\) is a Cantor set if all the critical points do escape to infinity under iteration. The authors give a complete solution of the problem when the Julia set of a cubic polynomial \(P\) is a Cantor set: The Julia set \(J_ P\) is a Cantor set if and only if the critical component \(K_ P(c)\) is not periodic. In the chapters 1-4 the theory of patterns is built and \(K_ P(c)\) is defined in terms of this theory. Very briefly: The pattern is a finite or infinite sequence of recursively determined Riemann surfaces. The notion of the end of patterns is defined in such a way, that the components of the filled-in Julia set \(K_ P\) for the polynomial \(P\) are in \(1-1\) correspondence with these ends. The component \(K_ P(c)\) corresponding to the critical end \(c\) is called the critical component. The chapters 7-11 are devoted to study so called parapatterns-parameter space for patterns. The last chapter deals with the polynomials of higher degree and this chapter says that most of the constructions of this paper go over to polynomials with two critical points of arbitrary degree. This paper demonstrates the beauty of complex analysis.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 665424
zbMATH Keywords: fundamental group
zbMATH Keywords: monodromy
zbMATH Keywords: critical points
zbMATH Keywords: Julia set
zbMATH Keywords: Cantor set
zbMATH Keywords: Riemann surfaces
reviewed by: A. Klíč
MaRDI profile type: Publication
cites work: Q5179675
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full work available at URL: https://doi.org/10.1007/bf02392761
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description: scientific article; zbMATH DE number 665424