Small sets of divergence points are dimensionless
The framework of this paper is as follows: one denotes by \(\Sigma\) a subshift modeled by a strongly connected graph, and by \(S:\Sigma\to\Sigma\) the associated shift. \({\mathcal P}(\Sigma)\) is the family of probability measures on \(\Sigma\). For \(n\in \mathbb N\), \(L_{n}:\Sigma\to{\mathcal P}(\Sigma)\) is the \(n\)th empirical measure, defined by \[ L_{n}\omega={{1}\over{n}}\sum_{k=0}^{n-1}\delta_{S^{k}\omega}, \] where \(\delta_{x}\) is the Dirac measure at \(x\). A continuous deformation of \(L_{n}\) is a pair \((X,\Xi)\), where \(X\) is a metric space, and \( \Xi:{\mathcal P}(\Sigma)\to X\) is a continuous map with respect to the weak topology. For two subsets \(T, U\) in \(X\) one defines \[ \Delta(T,U)=\{\omega\in\Sigma| T\subseteq A(\Xi L_{n} \omega)\subseteq U\}, \] where the symbol \(A\) denotes the set of accumulation points of the subsequent sequence. The author proves that the set \(\Delta(T,U)\) is dimensionless, i.e., it does not have an exact dimension function. More precisely, if \(t\) is its Hausdorff dimension and \(h\) is a dimension function of the form \(h(r)=r^{t}L(t)\), with \(L\) a slowly varying function, then if \(t<\text{dim} \Sigma\), we have \[ {\mathcal H}^{h}( \Delta(T,U))=0, \,\,\,\text{or}\,\,\, {\mathcal H}^{h}( \Delta(T,U))=\infty, \] in particular, \({\mathcal H}^{t}( \Delta(T,U))=0\) or \({\mathcal H}^{t}( \Delta(T,U))=\infty\). The main result is used to show that the multifractal decomposition sets of a general type of mixed multifractal spectrum of ergodic averages are dimensionless.
scientific article; zbMATH DE number 2051555