Hausdorff measure of sets of finite type of one-sided symbolic space
type: Item
instance of: scholarly article
title: Hausdorff measure of sets of finite type of one-sided symbolic space
zbMATH Open document ID: 0884.28007
DOI: 10.1007/BF02874518
author: Baoguo Jia
author: Zuoling Zhou
published in: Science in China. Series A
publication date: 1998-04-02T00:00:00Z
review text: Let \(A= (a_{ij})\) be a given \(k\times k\) \(0,1\)-matrix and \(\Sigma_A\) the space of all sequences \({\mathbf x}= (x_0,x_1,\dots)\) with \(x_i\in\{0,1,\dots, k-1\}\) such that \(a_{x_ix_{i+1}}= 1\) equipped with the metric \[ d({\mathbf x},{\mathbf y})= \begin{cases} 0 & \text{if}\quad{\mathbf x}={\mathbf y}\\ {1\over k^N} & \text{if}\quad{\mathbf x}\neq {\mathbf y},\;N=\min\{n\mid x_n\neq y_n\}.\end{cases} \] The Hausdorff dimension of \(\Sigma_A\) is well known to be \[ s=\dim_H(\Sigma_A)= {\log\rho(A)\over \log k}. \] The authors estimate the Hausdorff measure \(H^s(\Sigma_A)\) in two cases: 1) If \(A\) is aperiodic then lower and upper bounds can be expressed in terms of the spectral radius \(\rho(A)\) and the corresponding eigenvectors. 2) If \(A\) satisfies the condition of refinement invariant then for \(H^s(\Sigma_A)\) an exact formula is given.
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1066819
zbMATH Keywords: finite subshift
zbMATH Keywords: one-sided symbolic space
zbMATH Keywords: aperiodic
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Hausdorff measure
reviewed by: Hermann Haase
MaRDI profile type: Publication
cites work: Intersecting random translates of invariant Cantor sets
cites work: Q3943082
cites work: Q3931654
cites work: The topological Markov chain—Transitivity and mixing
cites work: Q3995301
full work available at URL: https://doi.org/10.1007/bf02874518
OpenAlex ID: W1994953995
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description: scientific article; zbMATH DE number 1066819