Cantor bouquets, explosions, and Knaster continua: Dynamics of complex exponentials
type: Item
instance of: scholarly article
title: Cantor bouquets, explosions, and Knaster continua: Dynamics of complex exponentials
zbMATH Open document ID: 0939.37028
DOI: 10.5565/PUBLMAT_43199_02
author: Robert L. Devaney
published in: Publicacions Matemàtiques
publication date: 2000-04-03T00:00:00Z
full work available at URL: https://eudml.org/doc/41369
review text: This paper is concerned with the topology and dynamics of the complex exponential family \(E_\lambda(z)= \lambda\exp^z\) with a real and positive parameter \(\lambda\). In the Julia set, \(J(E_\lambda)\), of \(E_\lambda\) there are two interesting topological structures: Cantor bouquets and Knaster-like continua. The author describes the construction of each in detail. As the parameter \(\lambda\) passes through the value \(1/e\), the Julia set \(J(E_\lambda)\) undergoes a remarkable transformations. The author shows that \(\bullet\) For \(0\leq\lambda<1/e\), \(J(E_\lambda)\) is a Cantor bouquet, which is a nowhere dense subset of the right half plane. \(\bullet\) At \(\lambda= 1/e\), \(J(E_\lambda)\) undergoes a simple saddle-node bifurcation. At this bifurcation, the attracting fixed point and its entire attracting basin disappear. However, a new topological invariant set arises. \(\bullet\) For \(\lambda>1/e\), \(J(E_\lambda)\) becomes the whole plane. And this new set is a Knaster-like indecomposable continuum on which most orbits cycle toward the orbit of 0 and \(\infty\).
reviewed by: Kiyoko Nishizawa
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 30D99
zbMATH DE Number: 1331806
zbMATH Keywords: holomorphic family of dynamical systems
zbMATH Keywords: bifurcation
zbMATH Keywords: exponential function
zbMATH Keywords: Knaster continua
zbMATH Keywords: Julia set
zbMATH Keywords: Cantor bouquet
zbMATH Keywords: attracting fixed point
zbMATH Keywords: attracting basin
zbMATH Keywords: indecomposable continuum
zbMATH Keywords: cycle orbits
MaRDI profile type: Publication
OpenAlex ID: W2099796788
description: scientific article; zbMATH DE number 1331806