Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity.
It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy's program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization.
Nevertheless, recently Berthé, Steiner, and Thuswaldner (2016) made some further progress on Rauzy's program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982).
Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincaré section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin's idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well.
It is the aim of my series of lectures to review the above results.[-]

The aim of this lecture is to find good symbolic codings for translations on the $d$-dimensional torus that enjoy the well-known and nice properties of Sturmian sequences (as for instance low complexity and good local discrepancy properties, i.e., bounded remainder sets of any scale). Inspired by the approach of G. Rauzy we construct such codings by the use of multidimensional continued fraction algorithms that are realized by sequences of substitutions. This is joint work with V. Berthé and W. Steiner.[-]

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).[-]