Computing the Hausdorff dimension of subshifts using matrices
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.
scientific article; zbMATH DE number 1129384
scientific article