Thermodynamic formalism and variations of the Hausdorff dimension of quadratic Julia sets
type: Item
instance of: scholarly article
title: Thermodynamic formalism and variations of the Hausdorff dimension of quadratic Julia sets
zbMATH Open document ID: 0985.37043
author: Guillaume Havard
author: Michel Zinsmeister
published in: Communications in Mathematical Physics
publication date: 2000-05-10T00:00:00Z
review text: This paper deals with variations of the Hausdorff dimension of quadratic Julia sets. Let \(P_c\) be a quadratic family of polynomials \[ P_c: z\mapsto z^2+ c,\quad c\in\mathbb{C}\tag{1} \] and \(d(c)\) the Hausdorff dimension of the Julia sets of (1). The authors prove that \(c\mapsto d'(c)\) tends to \(+\infty\) from the left at \({1\over 4}\) as \(({1\over 4}- c)^{d({1\over 4})-{3\over 2}}\). Hence the authors determine the speed of convergence of \(d(c)\) towards \(d({1\over 4})\) from the left. In particular, the graph of \(d\) has a vertical tangent on the left at \({1\over 4}\), which supports the numerical experiment.
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1445819
zbMATH Keywords: Julia set
zbMATH Keywords: quadratic family
zbMATH Keywords: Hausdorff dimension
Wikidata QID: Q123022594
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: s002200050778
OpenAlex ID: W2065307815
DOI: 10.1007/S002200050778
description: scientific article; zbMATH DE number 1445819