Hausdorff measure of sets of finite type of one-sided symbolic space
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.
scientific article; zbMATH DE number 1066819
scientific article