Stéphane Seuret

A PURE JUMP MARKOV PROCESS WITH A RANDOM SINGULARITY SPECTRUM

We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.

Ubiquity and large intersections properties under digit frequencies constraints

We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n=1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.

Diophantine approximation by orbits of expanding Markov maps

Given a dynamical system ([0, 1], T ), the distribution properties of the orbits of real numbers x ∈ [0, 1] under T constitute a longstanding problem. In 1995, Hill and Velani introduced the ”shrinking targets” theory, which aims at investigating precisely the Hausdorff dimensions of sets whose orbits are close to some fixed point. In this paper, we study the sets of points well-approximated by orbits {Tx}n≥0, where T is an expanding Markov map with finite partitions supported by the whole interval [0, 1]. The values of the dimensions of sets of well-approximable points are described using the multifractal properties of Gibbs measures invariant under the action of T . This study can be viewed as a moving shrinking targets problem.

Renewal of singularity sets of random self-similar measures

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

Recent developments in Fractals and related Fields

Multifractional Brownian motion (mBm), denoted here by X, is one of the paradigmatic examples of a continuous Gaussian process whose pointwise Hölder exponent depends on the location. Recall that X can be obtained (see e.g. [BJR97, AT05]) by replacing the constant Hurst parameter H in the standard wavelet series representation of fractional Brownian motion (fBm) by a smooth function H(·) depending on the time variable t. Another natural idea (see [BBCI00]) which allows to construct a continuous Gaussian process, denoted by Z, whose pointwise Hölder exponent does not remain constant all along its trajectory, consists in substituting H(k/2) to H in each term of index (j, k) of the standard wavelet series representation of fBm. The main goal of our article is to show that X and Z only differ by a process R which is smoother than them; this means that they are very similar from a fractal geometry point of view.Geometric Measure Theory and Multifractals.- Occupation Measure and Level Sets of the Weierstrass-Cellerier Function.- Space-Filling Functions and Davenport Series.- Dimensions and Porosities.- On Upper Conical Density Results.- On the Dimension of Iterated Sumsets.- Geometric Measures for Fractals.- Harmonic and Functional Analysis and Signal Processing..- A Walk from Multifractal Analysis to Functional Analysis with Spaces, and Back.- Concentration of the Integral Norm of Idempotents.- Le calcul symbolique dans certaines algebres de type Sobolev.- Lp-Norms and Fractal Dimensions of Continuous Function Graphs.- Uncertainty Principles, Prolate Spheroidal Wave Functions, and Applications.- 2-Microlocal Besov Spaces.- Refraction on Multilayers.- Wavelet Shrinkage: From Sparsity and Robust Testing to Smooth Adaptation.- Dynamical Systems and Analysis on Fractals..- Simple Infinitely Ramified Self-Similar Sets.- Quantitative Uniform Hitting in Exponentially Mixing Systems.- Some Remarks on the Hausdorff and Spectral Dimension of V-Variable Nested Fractals.- Cantor Boundary Behavior of Analytic Functions.- Measures of Full Dimension on Self-Affine Graphs.- Stochastic Processes and Random Fractals.- A Process Very Similar to Multifractional Brownian Motion.- Gaussian Fields Satisfying Simultaneous Operator Scaling Relations.- On Randomly Placed Arcs on the Circle.- T-Martingales, Size Biasing, and Tree Polymer Cascades.- Combinatorics on Words.- Univoque Numbers and Automatic Sequences.- A Crash Look into Applications of Aperiodic Substitutive Sequences.- Invertible Substitutions with a Common Periodic Point.- Some Studies on Markov-Type Equations.

Typical Borel measures on [0, 1]d satisfy a multifractal formalism

In this paper, we prove that in the Baire category sense, measures supported by the unit cube of typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures ?. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension d at exponent d, and it indeed coincides with the Legendre transform of the Lq-spectrum associated with typical measures ?.

Measures and functions with prescribed homogeneous multifractal spectrum

In this paper we construct measures supported in

Multifractal Analysis and Wavelets

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The singularity spectrum of the inverse of cookie-cutters

with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of

Ultrasonic characterization and multiscale analysis for the evaluation of dental implant stability: A sensitivity study

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