On the connectedness of limit net sets
type: Item
instance of: scholarly article
title: On the connectedness of limit net sets
zbMATH Open document ID: 1154.54307
author: B. E. Eshmatov
published in: Topology and its Applications
publication date: 2008-09-23T00:00:00Z
review text: The fractal objects considered in this paper are limit net sets described by means of net matrices (see definition below). The limit net sets can also be viewed as the limit set of a Moran construction (subsection 3.3) and from some known results on Moran fractals follows that limit net sets are fractals whose Hausdorff and box-counting dimensions equal \(1+\log 3/\log 4\) . More generally, limit net sets are also a species of random fractals, where the self-similarity property is replaced by statistical self-similarity, that means that enlargements of small parts have the same statistical distribution as the whole set (it can be checked that limit net sets are a particular case of random net fractals introduced by \textit{K. J. Falconer} in [Math. Proc. Camb. Philos. Soc. 100, 559--582 (1986; Zbl 0623.60020)]; see also [\textit{K. J. Falconer}, Fractal Geometry, Mathematical Foundations and Applications. Chichester etc.: John Wiley \& Sons (1990; Zbl 0689.28003)]. First we need to explain some definitions considered in the paper. A matrix \( A_{0}=(a_{i,j}^{0})_{0\leq i,j\leq 3}\) with entries in \(\{0,1\}\) is a net matrix of level \(0\) if exactly \(4\) entries equal \(1\), in each row and in each column of \(A\) there is exactly one entry with value \(1\) and in each submatrix \((a_{i,j})_{i=k,k+1,j=l,l+1}\), where \(k,l\in \{0,2\}\), there is exactly one entry with value \(1\). There are \(16\) net matrices. Two entries of a net matrix are neighbours if either their column indices coincide and their row indices are consecutive integers or their row indices coincide and their column indices are consecutive integers. A path in a net matrix is a finite sequence of neighbouring \(0\) entries of the matrix. A net matrix is said to be connected if for any two \(0\) entries there exists a path in the matrix that connects them. If we substitute in the net matrix \(A_{0}=(a_{i,j}^{0})_{0\leq i,j\leq 3}\) of level \(0\) each entry \(0\) by one net matrix and each entry \…
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 5346284
zbMATH Keywords: Connectedness
zbMATH Keywords: Moran fractal
zbMATH Keywords: Random fractals
reviewed by: M. Pellicer
MaRDI profile type: Publication
full work available at URL: j.topol.2008.06.001
OpenAlex ID: W1990735088
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DOI: 10.1016/J.TOPOL.2008.06.001
description: scientific article; zbMATH DE number 5346284