The talk will present some recent advances at the crossroads between Number Theory and Fractal Geometry requiring the input of algebraic theories to estimate the measure and/or the factal dimension of sets emerging naturally in Diophantine Approximation. Examples include the proof of metric, uniform and quantitative versions of the Oppenheim conjecture generalised to the case of any homogeneous form and also the determination of the Hausdor dimension of the set of well approximable points lying on polynomially dened manifolds (i.e. on algebraic varieties).[-]

We consider the monogenic representation for self-similar random fields. This approach is based on the monogenic representation of a greyscale image, using Riesz transform, and is particularly well-adapted to detect directionality of self-similar Gaussian fields. In particular, we focus on distributions of monogenic parameters defined as amplitude, orientation and phase of the spherical coordinates of the wavelet monogenic representation. This allows us to define estimators for some anisotropic fractional fields. We then consider the elliptical monogenic model to define vector-valued random fields according to natural colors, using the RGB color model. Joint work with Philippe Carre (XLIM, Poitiers), Céline Lacaux (LMA, Avignon) and Claire Launay (IDP, Tours).[-]

We study skew products of circle diffeomorphisms over a shift space. Our primary motivation is the fact that they capture some key mechanisms of nonhyperbolic behavior of robustly transitive dynamical systems. We perform a multifractal analysis of fiber-Lyapunov exponents studying the topological entropy of fibers with equal exponent. This includes the study of restricted variational principles of the entropy of ergodic measures with given fiber-exponent, in particular, with exponent zero. This enables to understand transitive dynamical systems in which hyperbolicities of different type are intermingled. Moreover, it enables to 'quantify of the amount of non-hyperbolicity' in a context where any other tools presently available fail. This is joint work with L.J. Díaz and M. Rams.[-]

In this talk, we adopt the viewpoint about fractional fields which is given in Lodhia and al. Fractional Gaussian fields: a survey, Probab. Surv. 13 (2016), 1-56. As example, we focus on random fields defined on the Sierpiński gasket but random fields defined on fractional metric spaces can also be considered. Hence, for $s \geq 0$, we consider the random measure $X=(-\Delta)^{-s} W$ where $\Delta$ is a Laplacian on the Sierpiński gasket $K$ equipped with its Hausdorff measure $\mu$ and where $W$ is a Gaussian random measure with intensity $\mu$. For a range of values of the parameter $s$, the random measure $X$ admits a Gaussian random field $(X(x))_{x \in K}$ as density with respect to $\mu$. Moreover, using entropy method, an upper bound of the modulus of continuity of $(X(x))_{x \in K}$ is obtained, which leads to the existence of a modification with Hölder sample paths. Along the way we prove sharp global Hölder regularity estimates for the fractional Riesz kernels on the gasket. In addition, the fractional Gaussian random field $X$ is invariant by the symmetries of the gasket. If time allows, some extension to $\alpha$-stable random fields will also be presented. Especially, for $s \geq s_0$ there still exists a modification of the $\alpha$-stable field $\mathrm{X}$ with Hölder sample paths whereas for $s< s_{0}$, such modification does not exist. This is a joint work with Fabrice Baudoin (University of Connecticut).[-]

Multifractal properties of data coming from many scientific fields (especially in turbulence) are now rigorously established. Unfortunately, the parameters measured on these data do not correspond to those mathematically obtained for the typical (or almost sure) functions in the standard functional spaces: Hölder, Sobolev, Besov…
In this talk, we introduce very natural Besov spaces in which typical functions possess very rich scaling properties, mimicking those observed on data for instance. We obtain various characterizations of these function spaces, in terms of oscillations or wavelet coefficients.
Combining this with the construction of almost-doubling measures with prescribed scaling properties, we are able to bring a solution to the so-called Frisch-Parisi conjecture. This is a joint work with Julien Barral (Université Paris-Nord).[-]