Dimension spectra of self-affine sets
type: Item
instance of: scholarly article
title: Dimension spectra of self-affine sets
zbMATH Open document ID: 1011.28004
DOI: 10.1007/BF02784523
author: Satoshi Takahashi
published in: Israel Journal of Mathematics
publication date: 2003-05-29T00:00:00Z
review text: A function \(H(\delta)\) -- the dimension spectrum -- monitoring the dimensions of sections of a self-affine carpet \(X\) is considered. Specifically, \(H(\delta)\) is defined as the Hausdorff dimension of the set of vertical coordinates that correspond to horizontal sections of \(X\) with box-counting (or Hausdorff) dimension \(\delta\). If \(X\) is irreducible, a version of the multifractal formalism is obtained in which \(H(\delta)\) coincides with the (modified) Legendre transform of a suitable free energy associated with \(H(\delta)\). The proof uses the fact that the multifractal formalism holds for irreducible sofic measures, which is proved by the author elsewhere. A variational formula is then derived for the Hausdorff dimension of \(X\), namely \(\dim_{\text{H}} X=\max_{\delta}\{\delta +H(\delta)\}\), which is also expressed in terms of the free energy.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1786371
zbMATH Keywords: self-affine carpets
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: box-counting dimension
zbMATH Keywords: sofic measures
zbMATH Keywords: multifractals
MaRDI profile type: Publication
cites work: Q5515914
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description: scientific article; zbMATH DE number 1786371