On Hausdorff dimension of Julia sets of hyperbolic rational semigroups
type: Item
instance of: scholarly article
title: On Hausdorff dimension of Julia sets of hyperbolic rational semigroups
zbMATH Open document ID: 0917.58038
author: Hiroki Sumi
published in: Kodai Mathematical Journal
publication date: 1999-08-05T00:00:00Z
review text: Let \(G\) be a finitely generated semigroup of rational maps of the Riemann sphere to itself. The purpose of this paper is first to investigate various notions of hyperbolicity of the semigroup. The author first shows that hyperbolicity is equivalent to being expanding and he introduces the stronger notion of ``the strong open set condition''. At the end of the paper, he constructs a sequence of examples satisfying this condition. Assuming it, the author shows first that the Lebesgue measure of the Julia set of \(G\) is 0, and, under a minor additional condition, shows that not only does the Hausdorff measure lie between 0 and 2 but that the Hausdorff measure at this dimension is finite. Under the assumption that \(G\) is expanding, the author gives an upper bound for the Hausdorff dimension of the Julia set.
Mathematics Subject Classification ID: 37F99
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1208192
zbMATH Keywords: rational semigroup
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: expanding
zbMATH Keywords: Julia set
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
OpenAlex ID: W2074641402
Wikidata QID: Q122516311
cites work: Q4026042
cites work: The Dynamics of Semigroups of Rational Functions I
cites work: Julia sets of rational semigroups
cites work: Complex Dynamics and Renormalization (AM-135)
cites work: The limit set of a Fuchsian group
cites work: The Julia sets of the random iteration of rational functions
cites work: Q4388655
cites work: Q3036339
cites work: Q4252537
cites work: On dynamics of hyperbolic rational semigroups
cites work: Pointwise Hölder exponents of the complex analogues of the Takagi function in random complex dynamics
cites work: Q4391301
cites work: Q4703483
DOI: 10.2996/KMJ/1138043831
description: scientific article; zbMATH DE number 1208192