Threshold and Hausdorff Spectrum of Discontinuous Measures

Abstract

Let $ m $ be a finite Borel measure on $[0,1]^d$. Consider the $L^q$-spectrum of $ m $: $\tau_ m (q)=\liminf_{n\to\infty}-n^{-1}\log_b\sum_{Q\in\mathcal{G}_n,\ m (Q)\neq 0} m (Q)^q$, where $\mathcal{G}_n$ is the set of $b$-adic cubes of generation $n$. Let $q_\tau=\inf\{q\: \tau_ m (q)=0\}$ and $H_\tau=\tau_ m '(q_\tau^-)$. When $ m $ is a mono-dimensional continuous measure of information dimension $D$, $(q_\tau,H_\tau)=(1,D)$. When $ m $ is purely discontinuous, its information dimension is $D=0$, but the non-trivial pair $(q_\tau,H_\tau)$ may contain relevant information on the distribution of $ m $. The connection between $(q_\tau,H_\tau)$ and the large deviation spectrum of $ m $ is studied in a companion paper. This paper shows that when a discontinuous measure $ m $ possesses self-similarity properties, the pair $(q_\tau,H_\tau)$ may store the main multifractal properties of $ m $, in particular the Hausdorff spectrum. This is observed thanks to a threshold performed on~$ m $.