Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set
type: Item
instance of: scholarly article
title: Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set
zbMATH Open document ID: 0984.37047
DOI: 10.1007/BF02921810
author: Dan Erik Krarup Sørensen
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: The author presents two methods for producing connected but non-locally connected Julia sets \(J_c\) of infinitely renormalizable quadratic polynomials \(P_c(z)=z^2+c\) for \(c \in \mathbb C\). The main tool for the construction are parabolic perturbations which are defined as follows. Given a root \(c_1\) of a hyperbolic component \(H_1\) of the Mandelbrot set \(M\) such that \(H_1\) is a satellite of another hyperbolic component \(H_0\) of \(M\). Then a first order satellite type parabolic perturbation of \(c_1\) is a perturbation to some special nearby root \(c_2 \in H_1\) of a hyperbolic component \(H_2\). Now, an infinite order satellite type parabolic perturbation \(c_\infty\) is a limit of an infinite sequence of first order satellite type parabolic perturbations. If the boundary of a satellite is parametrized by the internal arguments it carries an orientation. Therefore, it is possible to distinguish perturbations to the left and to the right. An infinite order parabolic perturbation is called one-sided if all first order perturbations are either to the left or to the right, and it is called two-sided if the perturbations are alternating. In the one-sided case the author arranges the perturbations in such a way that the Julia set of the limiting parameter has an external ray which accumulates at two distinct symmetric points. Then one may think of the Julia set as containing a ``topologists spiral''. For two-sided perturbations the author obtains the weaker result that the impression of the interesting external ray contains two symmetric points but accumulation can be guaranteed only at one of them. Then one may have a ``topologists sine'' in mind. Let \(\alpha(c)\) denote the least repelling fixed point of \(P_c\), and let \(\beta(c)\) be the other fixed point. A spine of \(P_c\) is an arc in the filled Julia set of \(P_c\) joining \(\beta(c)\) and \(-\beta(c)\). In a more general setting, containing both of the above cases, the author proves the existence…
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F45
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F25
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 1542068
zbMATH Keywords: quadratic polynomial
zbMATH Keywords: Julia set
zbMATH Keywords: Mandelbrot set
zbMATH Keywords: local connectivity
zbMATH Keywords: accumulation set
zbMATH Keywords: prime end
zbMATH Keywords: impression
zbMATH Keywords: external ray
zbMATH Keywords: tuning
zbMATH Keywords: satellite
zbMATH Keywords: wake
zbMATH Keywords: infinitely renormalizable
zbMATH Keywords: hyperbolic components
zbMATH Keywords: robustness
Wikidata QID: Q121611739
MaRDI profile type: Publication
cites work: Q4177796
cites work: On the dynamics of polynomial-like mappings
cites work: Q3739420
cites work: Q4328166
cites work: Fixed points of polynomial maps. Part II. Fixed point portraits
cites work: Building blocks for quadratic Julia sets
cites work: Q4288672
cites work: Disconnected Julia set and rotation sets
cites work: Q4000499
cites work: Accumulation theorems for quadratic polynomials
cites work: Describing quadratic Cremer point polynomials by parabolic perturbations
description: scientific article; zbMATH DE number 1542068