On dynamics of hyperbolic rational semigroups
type: Item
instance of: scholarly article
title: On dynamics of hyperbolic rational semigroups
zbMATH Open document ID: 0938.37026
author: Hiroki Sumi
published in: Journal of Mathematics of Kyoto University
publication date: 2000-06-28T00:00:00Z
review text: The study of rational semigroups is a generalization of the study of Kleinian groups, iteration of rational functions and systems of contraction maps related to self-similar sets in \(\mathbb{C}\) in fractal geometry. In the framework of the study of rational semigroups, the author shows some basic results similar to those between Kleinian groups and iteration of rational functions. The author proves that if the hyperbolic rational semigroup is finitely generated and satisfies some natural conditions, the limit functions take their values on a postcritical set. Moreover, the author proves that when the generators of a finitely generated hyperbolic rational semigroup are perturbed, the hyperbolicity is kept and the Julia set depends continuously on the generators of the semigroup. Furthermore, if the finitely generated rational semigroup is hyperbolic and if the inverse images by the generators of the Julia set are mutually disjoint, then the Julia set moves by holomorphic motion.
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 1295314
zbMATH Keywords: rational semigroups
zbMATH Keywords: fractal geometry
zbMATH Keywords: postcritical set
zbMATH Keywords: Julia set
zbMATH Keywords: hyperbolicity
zbMATH Keywords: Kleinian groups
zbMATH Keywords: hyperbolic rational semigroup
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518211
OpenAlex ID: W1646822462
DOI: 10.1215/KJM/1250518211
description: scientific article; zbMATH DE number 1295314