Computing the Hausdorff dimension of subshifts using matrices
type: Item
instance of: scholarly article
title: Computing the Hausdorff dimension of subshifts using matrices
zbMATH Open document ID: 0893.58023
DOI: 10.1016/S0024-3795(97)00371-6
author: S. Friedland
published in: Linear Algebra and its Applications
publication date: 1998-07-26T00:00:00Z
review text: For a one-sided subshift of finite type, the author considers a class of pseudometrics associated to nonnegative weights \(w\) on the edges of graphs presenting them. The distance between two sequences \((a_i),(b_i)\) is essentially the product \(w(a_1,a_2)\cdots w(a_k,a_{k+1})\) of the weights on edges \(a_i,a_{i+1}\) such that \(a_1=b_1,\ldots, a_{k+1}=b_{k+1}\). The Hausdorff dimension of this metric space is computable by a formula of Mauldin-Williams, \((\log \rho( \Delta ))/h\) provided that \(b_{ij}=e^{-hm_{ij}}\) for nonnegative integers \(m\), where \(\rho \) is the spectral radius of an associated graph which between vertices \(i,j\) has an edge sequence of length \(m_{ij}\) disjoint from other such edge sequences. This is shown to apply to a natural metric for free actions of groups on locally finite trees, and then to give a measure on subgroups of finitely generated free groups, and applied specifically to finitely generated free subgroups of the modular group. If \(G\subset H\), then their measures are equal if and only if the index of \(G\) in \(H\) is finite.
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 1129384
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: subshift of finite type
zbMATH Keywords: modular group
zbMATH Keywords: subgroups of free groups
zbMATH Keywords: finitely generated free subgroups
reviewed by: Ki Hang Kim
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1129384