Escape rates and Hausdorff dimension of Julia sets
type: Item
instance of: scholarly article
title: Escape rates and Hausdorff dimension of Julia sets
zbMATH Open document ID: 0787.58036
DOI: 10.1016/0960-0779(93)90023-T
author: Bruno Eckhardt
published in: Chaos, Solitons and Fractals
publication date: 1993-11-01T00:00:00Z
review text: A periodic orbit formalism is used to compute escape rate and Hausdorff dimension for the Julia sets of the one parameter family of maps \(z \Rightarrow z^ 2 + c\) in a neighborhood of \(c = 0\). The method is based on \(\Omega(z,D) = \sum^ \infty_{n = 1}z^ n\sum_ P|\Lambda_ P|^{-D}\), where \(\Lambda_ P\) is the derivative of the \(n\)-th iterate of the function at the point \(P\) of the complex plane, periodic after \(n\) iterations. \(\Omega(z,D)\) can be viewed as a logarithmic derivative of a zeta function and so the escape rate and Hausdorff dimension is calculated as certain zeros of this zeta function.
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 427741
zbMATH Keywords: Julia sets
zbMATH Keywords: zeta function
zbMATH Keywords: escape rate
zbMATH Keywords: Hausdorff dimension
reviewed by: László Máté
MaRDI profile type: Publication
Wikidata QID: Q62599045
cites work: Q4740120
cites work: Q3736578
cites work: Repellers for real analytic maps
cites work: Strange objects in the complex plane
cites work: Thermodynamic Formalism
full work available at URL: 0960-0779(93)90023-t
OpenAlex ID: W2042350246
description: scientific article; zbMATH DE number 427741