The finer geometry and dynamics of the hyperbolic exponential family
type: Item
instance of: scholarly article
title: The finer geometry and dynamics of the hyperbolic exponential family
zbMATH Open document ID: 1038.37037
author: Mariusz Urbański
author: Anna Zdunik
published in: Michigan Mathematical Journal
publication date: 2003-10-27T00:00:00Z
review text: The paper contains a study of the maps \(f_\lambda(z) = \lambda e^z\), where \(\lambda\) is a nonzero complex number. It concentrates on the subset \(J_r(f_\lambda)\) of the Julia set of \(f_\lambda\) consisting of points whose forward iterates by \(f_\lambda\) do not go to infinity. A detailed proof is given that, if \(0<\lambda < 1/e\), then the Hausdorff dimension of \(J_r(f_\lambda)\) is less than 2. The importance of the restriction on \(\lambda\) is that it ensures that \(f_\lambda\) has an attracting fixed-point. The changes necessary to extend the result to \(f_\lambda\) with an attracting periodic orbit are outlined. The argument goes by using \(f_\lambda\) to induce a map \(F\) from some horizontal strip of height \(2\pi i\) to itself (using the fact that \(f_\lambda\) has period \(2\pi i\)). It is shown that there is a unique conformal probability measure for \(F\) and a \(\sigma\)-finite conformal measure for \(F_\lambda\). A recent result of M. Martens is used to show the existence of a unique ergodic Borel probability measure for \(F\) that is equivalent with the conformal measure.
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 37F35
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 1997311
zbMATH Keywords: Julia set
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: attracting fixed-point
zbMATH Keywords: attracting periodic orbit
zbMATH Keywords: conformal measure
reviewed by: Mike Hurley
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1060013195
OpenAlex ID: W2046490002
cites work: Area distortion of quasiconformal mappings
cites work: Tying hairs for structurally stable exponentials
cites work: Hausdorff dimension of quasi-circles
cites work: On the Existence of Conformal Measures
cites work: On Sullivan's conformal measures for rational maps of the Riemann sphere
cites work: Geometric measures for parabolic rational maps
cites work: Q5565773
cites work: Q4939491
cites work: Q4097944
cites work: Q4248086
cites work: Q5685358
cites work: Area and Hausdorff Dimension of Julia Sets of Entire Functions
cites work: Q3563981
cites work: Packing Measure, and its Evaluation for a Brownian Path
DOI: 10.1307/MMJ/1060013195
description: scientific article; zbMATH DE number 1997311