An introduction to Mandelbrot cascades

In this course, we propose an elementary and self-contained introduction to canonical Mandelbrot random cascades. The multiplicative construction is explained and the necessary and sufficient condition of non-degeneracy is proved. Then, we discuss the problem of the existence of moments and the link with nondegeneracy. We also calculate the almost sure dimension of the measures. Finally, we give an outline on multifractal analysis of Mandelbrot cascades. This course was delivered in september 2013 during a meeting of the "Multifractal Analysis GDR" (GDR no 3475 of the french CNRS).Comment: 36 page

On measures driven by Markov chains

We study measures on $[0,1]$ which are driven by a finite Markov chain and which generalize the famous Bernoulli products. We propose a hands-on approach to determine the structure function $\tau$ and to prove that the multifractal formalism is satisfied. Formulas for the dimension of the measures and for the Hausdorff dimension of their supports are also provided

Measures and the Law of the Iterated Logarithm

Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this question in several situations (self-similar measures, quasi-Bernoulli measures). More precisely we obtain fine comparisons between the mesure m and generalized Hausdorff type (or packing type) measures. The Law of the Iterated Logarithm or estimations of the L^q-spectrum in a neighborhood of q=1 are the tools to obtain such results.Comment: 18 page

Multifractal analysis of images: New connexions between analysis and geometry

Natural images can be modelled as patchworks of homogeneous textures with rough contours. The following stages play a key role in their analysis: - Separation of each component - Characterization of the corresponding textures - Determination of the geometric properties of their contours. Multifractal analysis proposes to classify functions by using as relevant parameters the dimensions of their sets of singularities. This framework can be used as a classification tool in the last two steps enumerated above. Several variants of multifractal analysis were introduced, depending on the notion of singularity which is used. We describe the variants based on Hölder and L^p regularity, and we apply these notions to the study of functions of bounded variation (indeed the BV setting is a standard functional assumption for modelling images, which is currently used in the first step for instance). We also develop a multifractal analysis adapted to contours, where the regularity exponent associated with points of the boundary is based on an accessibility condition. Its purpose is to supply classification tools for domains with fractal boundaries

Multifractal analysis of the divergence of Fourier series: the extreme cases

International audienceWe study the size, in terms of the Hausdorff dimension, of the subsets of $\mathbb T$ such that the Fourier series of a generic function in L^1(\TT), L^p(\TT) or in $\mathcal C(\mathbb T)$ may behave badly. Genericity is related to the Baire category theorem or to the notion of prevalence

MULTIFRACTAL PHENOMENA AND PACKING DIMENSION

We undertake a general study of multifractal phenomena for functions. We show that the existence of several kinds of multifractal functions can be easily deduced from an abstract statement, leading to new results. This general approach does not work for Fourier or Dirichlet series. Using careful constructions, we extend our results to these cases

On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets

Let $K$ be a compact set in \rd with positive Hausdorff dimension. Using a Fractional Brownian Motion, we prove that in a prevalent set of continuous functions on $K$, the Hausdorff dimension of the graph is equal to $\dim_{\mathcal H}(K)+1$. This is the largest possible value. This result generalizes a previous work due to J.M. Fraser and J.T. Hyde which was exposed in the conference {\it Fractal and Related Fields~2}. The case of $\alpha$-H\"olderian functions is also discussed

Boundary multifractal behaviour for harmonic functions in the ball

It is well known that if $h$ is a nonnegative harmonic function in the ball of \RR^{d+1} or if $h$ is harmonic in the ball with integrable boundary values, then the radial limit of $h$ exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any $\beta\in [0,d]$, the Hausdorff dimension of the set of points $\xi$ on the sphere such that $h(r\xi)$ looks like $(1-r)^{-\beta}$ is equal to $d-\beta$.Comment: 16 page