The shape of the Julia set of an expanding rational map
type: Item
instance of: scholarly article
title: The shape of the Julia set of an expanding rational map
zbMATH Open document ID: 1395.54014
author: José Ignacio Extremiana Aldana
author: L. Javier Hernández
author: Maria Teresa Rivas Rodriguez
published in: Topology and its Applications
publication date: 2018-03-27T00:00:00Z
review text: In this paper the authors establish a bridge between computational topology and shape/strong shape theory. The techniques developed permit to construct resolutions of CW complexes and to study the shape of the Julia set of a rational map (using topological, geometrical, and computational tools). The original contribution of this article is the construction of the shape resolutions using the spherical multiplier of an analytic function defined on the Riemann sphere. The authors implement an algorithm for computing a finite number of cubic complexes of these resolutions. This can be used to obtain global visualizations of the Julia set of an expanding rational map or to compute homotopical/homological shape invariants. A global graphic representation of the Julia set on the unit 2-sphere is obtained. The inverse system of cubical complexes is used to (i) obtain nice global visualizations of the fractal structure of the Julia set \(J(g)\) (ii) determine the shape of (the compact metric space) \(J(g)\). The main advantage of this algorithm based on spherical multipliers is that it is very fast. As further work the authors intend to implement in Julia language (high level/high performance dynamic programming language) the algorithms presented in this paper with possible main steps (iv)--(viii) presented in the final section. The paper also contains examples.
reviewed by: Corina Mohorianu
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 6853365
zbMATH Keywords: shape
zbMATH Keywords: Julia set
zbMATH Keywords: rational map
zbMATH Keywords: spherical multiplier
describes a project that uses: PBURF03.jl
describes a project that uses: Julia
MaRDI profile type: Publication
full work available at URL: j.topol.2018.02.028
OpenAlex ID: W2791726474
Wikidata QID: Q122718051
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DOI: 10.1016/J.TOPOL.2018.02.028
description: scientific article; zbMATH DE number 6853365