Graph-directed structures of self-similar sets with overlaps
type: Item
instance of: scholarly article
title: Graph-directed structures of self-similar sets with overlaps
zbMATH Open document ID: 0969.28003
DOI: 10.1142/S025295990000042X
author: Hui Rao
author: Hua Su
published in: Chinese Annals of Mathematics. Series B
publication date: 2001-09-30T00:00:00Z
review text: This paper shows that a certain family of self-similar sets of the real line (i.e., satisfying the set-theoretic equation \(K= \bigcup_{i= 1} ^{m} \varphi_i(K)\), each \(\varphi_i\) being a contractive similarity on \({\mathbf R}\)) with non-trivial overlapping can be obtained as invariant sets associated with a graph-directed self-similar construction satisfying the standard (``open set'') separation condition. The class under consideration includes as particular cases some \(\lambda\)-Cantor sets (i.e., \(m=3\) and \(\varphi_i(x)= x/3+a_i\), \(a_i=0,\lambda,2/3\) with \(\lambda\in (0,1/3)\)) and some sets associated with the so-called \(\{0,1,3\}\)-problem (i.e., \(m=3\) and \(\varphi_i(x)=\beta(x+i)\) with \(i=0,1,3\) and \(\beta\in (1/3,2/5)\)). Applying dimension theory for graph-directed self-similar sets, the fractional dimensions of some of these sets are computed.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1547186
zbMATH Keywords: self-similar sets
zbMATH Keywords: graph-directed self-similar sets
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: fractional dimensions
MaRDI profile type: Publication
full work available at URL: s025295990000042x
OpenAlex ID: W2396459477
description: scientific article; zbMATH DE number 1547186