Continuity of Julia set and its Hausdorff dimension of Yang-Lee zeros
type: Item
instance of: scholarly article
title: Continuity of Julia set and its Hausdorff dimension of Yang-Lee zeros
zbMATH Open document ID: 1222.37043
author: Junyang Gao
author: Jianyong Qiao
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-03-29T00:00:00Z
review text: In this paper, the authors studied the Julia set of the rational functions \[ S_{\lambda}(z)=\left(\frac{z^3+3(\lambda-1)z+(\lambda-1)(\lambda-2)}{3z^2+ 3(\lambda-2)z+\lambda^2-3\lambda+3}\right)^2, \] where \(\lambda\) is a real number. In fact, they proved that the Julia set \(J(S_{\lambda})\) of \(S_{\lambda}\) is discontinuous if and only if \(\lambda \in \{\alpha_0,0,\beta_0\},\) and the Hausdorff dimension \(HD(J(S_{\lambda}))\) of \(J(S_{\lambda})\) is continuous at \(\lambda\) if \(\lambda\in \mathbb{R}\setminus\{\alpha_0,0,\beta_0\},\) where \[ \alpha_0=\min_{1<t<2}\frac{(t^2-1)(\sqrt{4t+1}-3)}2,\quad \beta_0=\max_{0<t<1}\frac{(1-t^2)(\sqrt{4t+1}+3)}2. \]
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F35
zbMATH DE Number: 5871212
zbMATH Keywords: Junia set
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: continuity
zbMATH Keywords: renormalization transformations
reviewed by: Liangwen Liao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.01.016
OpenAlex ID: W2068323430
Wikidata QID: Q123010287
cites work: The components of a Julia set
cites work: Julia sets and complex singularities in hierarchical Ising models
cites work: Hausdorff measures on Julia sets of subexpanding rational maps
cites work: Q4328166
cites work: Hausdorff dimension and conformal dynamics. II: Geometrically finite rational maps
cites work: Dynamics in one complex variable. Introductory lectures.
cites work: Q3736578
cites work: On connectivity of Julia sets of Yang-Lee zeros
cites work: On Julia sets concerning phase transitions
cites work: On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets
cites work: Q3036339
cites work: Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation
cites work: Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model
DOI: 10.1016/J.JMAA.2011.01.016
description: scientific article; zbMATH DE number 5871212