Quartic Julia sets including any two copies of quadratic Julia sets
type: Item
instance of: scholarly article
title: Quartic Julia sets including any two copies of quadratic Julia sets
zbMATH Open document ID: 1370.37089
author: Koh Katagata
published in: Discrete and Continuous Dynamical Systems
publication date: 2015-10-27T00:00:00Z
review text: Let \(P\) be a polynomial of degree \(d\geqslant 2\) and denote by \(K(P)\) the filled-in Julia set of \(P\), which is the set of points that have bounded orbit under iteration by \(P\). It is well-known that if all finite critical points of \(P\) escape to infinity, then \(K(P)\) is totally disconnected, while \(K(P)\) is connected if and only if all finite critical points have a bounded orbit. This paper focuses on the remaining intermediate cases, where only some of the critical points escape to infinity, and hence \(K(P)\) is disconnected but not totally disconnected. In this situation, if we pick an equipotential curve surrounding \(K(P)\) such that it passes through an escaping critical value, there is a preimage \(\gamma\) of this curve that contains the critical point and has a figure-eight shape. The restriction of \(P\) to each bounded complementary component \(U\) of \(\gamma\) defines what is known as a polynomial-like map \((f,U,U')\) of degree \(d'<d\), where \(f=P|_{U}\) and \(U'=P(U)\). Then, one can define the filled-in Julia set of such map by considering the points that never leave \(U\), which is a subset of \(K(P)\). \textit{A. Douady} and \textit{J. H. Hubbard} [Ann. Sci. Éc. Norm. Supér. (4) 18, 287--343 (1985; Zbl 0587.30028)] introduced the notion of polynomial-like mapping to explain the appearance of sets resembling quadratic filled-in Julia sets in the dynamical planes of higher degree polynomials and transcendental entire functions, as well as copies of the Mandelbrot set in the parameter spaces of families of such functions. They showed that every polynomial-like map \(f\) is hybrid equivalent to a polynomial \(P\) of the same degree and, in particular, the filled Julia set of \(f\) is a quasiconformal image of \(K(P)\). This paper looks at the converse problem: given two filled-in Julia sets of quadratic polynomials \(P_1,P_2\), can we find a polynomial \(P\) of degree \(4\) such that \(K(P)\) contains quasiconformal copies of \(K(P_1)…
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 6498605
zbMATH Keywords: complex dynamics
zbMATH Keywords: Julia set
zbMATH Keywords: polynomial-like map
zbMATH Keywords: quasiconformal surgery
reviewed by: David Martí-Pete
MaRDI profile type: Publication
full work available at URL: dcds.2016.36.2103
OpenAlex ID: W2525225878
cites work: On the dynamics of polynomial-like mappings
cites work: On a certain kind of polynomials of degree 4 with disconnected Julia set
cites work: Q3530535
cites work: Q4933741
DOI: 10.3934/DCDS.2016.36.2103
description: scientific article; zbMATH DE number 6498605