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2006/2007 Threshold and Hausdorff Spectrum of Discontinuous Measures
Julien Barral, Stéphane Seuret
Real Anal. Exchange 32(2): 455-472 (2006/2007).


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 $.


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Julien Barral. Stéphane Seuret. "Threshold and Hausdorff Spectrum of Discontinuous Measures." Real Anal. Exchange 32 (2) 455 - 472, 2006/2007.


Published: 2006/2007
First available in Project Euclid: 3 January 2008

zbMATH: 1144.28003
MathSciNet: MR2369855

Primary: 28A80 , 60F10

Keywords: Fractal , large deviations

Rights: Copyright © 2006 Michigan State University Press

Vol.32 • No. 2 • 2006/2007
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