JCJC SIMI 1 - JCJC - SIMI 1 - Mathématiques et interactions

Multifractals and metric theory of Diophantine Approximation – MUTADIS

We have used all the classical methods in multifractal analysis, ergodic theory and probability,. We also have developed new methods based on large intersection properties and ubiquity theorems.

J. Barral supervise a PhD student since 2011 on multifractal analysis of discrete Gibbs measures.
J. Barral and S. Seuret have almost finished performing the multifractal analysis of a new percolation model on trees and on wavelet series weighted by Gibbs measures.
Together with S. Jaffard (Créteil), A. Durand has studied a higher dimensional model of Davenport series.
J. Barral, A. Durand and S. Seuret have together introduced a local multifractal formalism adpated to functions, measures and distributions (more generally, capacities) that are characterized by «moving« multifractal behavior, i.e. their spectrum may change from one region to the other.They gave several relevant examples of natural objects enjoying this property.
With Y. Bugeaud (Strasbourg), A. Durand focuses on Mahler's conjecture on well-approximated numbers in the Canto set. Based on probability results, they proposed a conjecture on the value of the Hausdorff dimension of these sets.
L. Liao worked with M. Rams (Varsovie) on new research on Diophantine approximation.
S. Seuret, joint with Y. Peres (IBM), J. Schmeling (Sweden) and B. Solomyak (Washington), computed the dimension of certain non-invariant sets naturally arising in dynamical systems, related with multiple ergodic averages (studied by Füstenberg, Bourgain, Host-Kra,...). On the same subject, L. Liao studied the averages in the case of an IFS with two liear branches.
S. Seuret, with T. Rivoal (DR Grenoble), worked on historical Fourier series and their convergence and multifractal properties. They found new functional equations for these series, and connected them to some non-standard continued fractions expansion.

To summarize the above, the members of the MUTADIS project have made great advances on the four research directions announced in the initial research plan.

- 2013: 10 articles in international journals (so far)
- 2012: 5 articles in international journals.

- Participation in 9 conferences (participation supported by the ANR project).

- Financial support and organization of two conferences.

Submission summary

In several domains (geophysics, cardiology, finance, and Internet traffic), numerical studies reveal that the fluctuations of characteristic quantities possess scaling invariance properties. These scaling invariance measured on empirical data are reminiscent of properties verified theoretically by mathematical objects, which are called (self-similar) multifractals. A multifractal is a measure, a function or a stochastic process, whose local regularity changes rapidly from one point to another. It is striking how multifractal properties arise in many mathematical fields: dynamical systems, probability, harmonic analysis, and recently Diophantine approximation. The project's main objective is the investigation of the various aspects of the interactions between multifractal theory (more generally, geometric measure theory) and metric number theory. Since the 1990's, such interactions appear to be key issues when performing the regularity analysis of many multifractal models, hence we also plan to develop many mathematical models (functions, measures, stochastic processes) based on the results we will obtain.

We will focus on four challenging directions:

- Multifractals and Diophantine approximation, Mass transference principles: Recently, interactions between multifractal theory and metric theory of Diophantine approximation have been pointed out, giving new perspectives in both fields. Examples are the heterogeneous Diophantine approximation results, which allow to investigate the approximation of real numbers by rational numbers which are constrained by the fact that their digit frequencies (in any basis) are fixed. We want to push further these interactions, in particular by studying heterogeneous approximation under other natural constraints, and by obtaining various mass transference principles in various contexts.

- Large intersection properties: A set E included in R^d belongs to the class of sets with large intersection properties (introduced by K. Falconer) when for any countable family of similarities of R^d, the Hausdorff dimension of the intersection of the sets f_i(E) has same Hausdorff dimension as E. This remarkable property has been proved to hold for many sets arising in metric number theory (in particular, there are many connections with the mass transference principles discussed in the preceding item), and plays an important role when performing the multifractal analysis of many objects. We aim at proving these properties for several sets families relevant in number theory, at introducing a notion of large intersection in spaces other than R^d (for instance inside a Cantor set), and at using our results for the study of the multifractal nature of various objects.

- Dynamical Diophantine Approximation: A longstanding issue concerns the equi-distribution properties of the orbit of a point x under the action of a dynamical system (X,T). Recently, there has been some remarkable progress for one of the most natural dynamical system: ([0,1], T) where T is the doubling shift, and the corresponding results on the distribution properties of the orbits rely on multifractal properties. Our goal is to obtain comparable results for more general dynamical systems (like expanding Markov maps, or the Gauss map). This will have consequences on number theory, since the Gauss map is intimately related to the continued fractions.

- Development of multifractal models: Since the 1990's, it has been regularly stated that the local regularity properties of many objects (Fourier or wavelet series, Lévy processes, "typical" functions) are related to questions of generalized Diophantine approximation, i.e. the approximation by families of points which are not the rational numbers. We intend to use the results we described above for two purposes: first to study the multifractal nature of classical objects, and then to develop new multifractal models (whose study requires new tools in metric number theory.

Project coordination

Stéphane SEURET (UNIVERSITE PARIS-EST CRETEIL VAL DE MARNE) – seuret@univ-paris12.fr

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.



Help of the ANR 60,000 euros
Beginning and duration of the scientific project: December 2011 - 48 Months

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