Inverse problems  in multifractal analysis of measures

Barral, Julien

doi:10.24033/asens.2274

Inverse problems in multifractal analysis of measures
[Problèmes inverses dans l'analyse multifractale des mesures]

Barral, Julien

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1457-1510.

Résumé
Abstract

Le formalisme multifractal est un cadre adapté pour décrire la distribution aux petites échelles des mesures de Borel finies positives à support compact dans $ℝ^{d}$ , dont l'ensemble est ici noté $ℳ_{c}^{+} (ℝ^{d})$ . Il est dit valide pour une mesure $μ$ lorsque son spectre de Hausdorff est la fonction semi-continue supérieurement obtenue comme transformée de Legendre-Fenchel concave de sa fonction d'énergie libre $τ_{μ}$ ; c'est le cas pour certaines classes fondamentales de mesures exactement dimensionnelles.

Pour toute fonction $τ$ candidate à être la fonction d'énergie libre d'un élément $μ$ de $ℳ_{c}^{+} (ℝ^{d})$ , nous construisons une telle mesure, exactement dimensionnelle, et validant le formalisme. Ce résultat s'étend à un formalisme plus fin considérant simultanément spectres de Hausdorff et de packing. D'autre part, pour toute fonction semi-continue supérieurement candidate à être le spectre de Hausdorff inférieur d'une mesure exactement dimensionnelle, nous construisons une telle mesure.

Multifractal formalism is designed to describe the distribution at small scales of the elements of $ℳ_{c}^{+} (ℝ^{d})$ , the set of positive, finite and compactly supported Borel measures on $ℝ^{d}$ . It is valid for such a measure $μ$ when its Hausdorff spectrum is the upper semi-continuous function given by the concave Legendre-Fenchel transform of the free energy function $τ_{μ}$ associated with $μ$ ; this is the case for fundamental classes of exactly dimensional measures.

For any function $τ$ candidate to be the free energy function of some $μ \in ℳ_{c}^{+} (ℝ^{d})$ , we construct such a measure, exactly dimensional, and obeying the multifractal formalism. This result is extended to a refined formalism considering jointly Hausdorff and packing spectra. Also, for any upper semi-continuous function candidate to be the lower Hausdorff spectrum of some exactly dimensional $μ \in ℳ_{c}^{+} (ℝ^{d})$ , we construct such a measure.

DOI : 10.24033/asens.2274

Classification : 28A78, 60F10
Keywords: Multifractal formalism, multifractal analysis, Hausdorff dimension, packing dimension, large deviations, inverse problems.
Mot clés : Formalisme multifractal, analyse multifractale, dimension de Hausdorff, dimension de packing, grandes déviations, problèmes inverses.

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Barral, Julien. Inverse problems  in multifractal analysis of measures. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1457-1510. doi : 10.24033/asens.2274. http://www.numdam.org/articles/10.24033/asens.2274/

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