An exact formula for the measure dimensions associated with a class of piecewise linear maps
type: Item
instance of: scholarly article
title: An exact formula for the measure dimensions associated with a class of piecewise linear maps
zbMATH Open document ID: 0666.28004
DOI: 10.1007/BF01889600
author: D. P. Hardin
author: Jeffrey S. Geronimo
published in: Constructive Approximation
publication date: 1989-01-01T00:00:00Z
review text: Let \(w_ 1,...,w_ N\) be similitudes contracting \(R^ n\) by the factors \(0<s_ i<1\), respectively. For given probabilities \(p_ 1,...,p_ N\) there exists a unique (``p-balanced'') measure \(\mu\) (supported on the unique compact set A satisfying \(A=\cup w_ iA)\) having the invariance property \(\mu =\sum p_ i\mu \circ w_ i^{-1}.\) Theorem 2. If \(w_ iA\cap w_ jA=\emptyset\) for \(i=j\), then both the Hausdorff-Besicovitch and the box dimensions of \(\mu\) equal \(\sum p_ i \log p_ i/\sum p_ i \log s_ i.\) For general affine maps \(w_ i\) the equality of both dimensions is no longer true (section 4).
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 4089838
zbMATH Keywords: self-similar fractals
zbMATH Keywords: Hausdorff-Besicovitch dimension
zbMATH Keywords: measure dimensions of attractors
zbMATH Keywords: box dimension
zbMATH Keywords: fractional dimension
zbMATH Keywords: similitudes
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01889600
OpenAlex ID: W4247786464
description: scientific article; zbMATH DE number 4089838