Hausdorff dimension of the arithmetic sum of self-similar sets
type: Item
instance of: scholarly article
title: Hausdorff dimension of the arithmetic sum of self-similar sets
zbMATH Open document ID: 1376.28005
author: Kan Jiang
published in: Indagationes Mathematicae. New Series
publication date: 2016-06-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.0505
review text: In his famous paper [Indiana Univ. Math. J. 30, 713--747 (1981; Zbl 0598.28011)] \textit{J. E. Hutchinson} proved that for each iterated function system (IFS) composed by self contractions on a complete metric space, there exists a unique invariant nonempty compact set, called the attractor. This is, generally, a fractal set. If, further, the contractions are similitudes, the attractor is a self-set and Hutchinson showed how its Hausdorff dimension can be obtained. In the present paper, the author considers the arithmetic sum of two self-similar sets generated by two iterated function systems composed by similitudes and he proves that this set is either a self-similar set or a unique attractor of some infinite iterated function system (Theorem 2.15). Moreover, under some conditions, the author evaluates the Hausdorff dimension of that sum.
reviewed by: Nicolae-Adrian Secelean
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37F35
zbMATH DE Number: 6595322
zbMATH Keywords: fractals
zbMATH Keywords: iterated function system
zbMATH Keywords: infinite iterated function system
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: self-similar set
zbMATH Keywords: arithmetic sum
MaRDI profile type: Publication
OpenAlex ID: W2963471676
arXiv ID: 1411.0505
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DOI: 10.1016/J.INDAG.2016.01.003
description: scientific article; zbMATH DE number 6595322