There is a classical connection between Riesz-potentials, Riesz-energies and Hausdorff dimension. Otto Frostman (Lund) proved in his thesis that if $E$ is a set and $\mu$ is a measure with support in $E$, then the Hausdorff dimension of $E$ is at least $s$ if the $s$-dimensional Riesz-energy of $\mu$ is finite. I will first recall Frostman’s result and some of its applications. I will then mention some new methods where Hausdorff dimension is calculated using potentials and energies with inhomogeneous kernels. Some applications are in stochastic geometry and dynamical systems.

Nous proposons dans cet exposé, la dérivation d'un modèle hydrodynamique 2D décrivant un disque d'accrétion en astrophysique à partir de la formulation 3D.
Il s'agit d'une application de la procédure de réduction dimensionnelle introduite récemment par Maltese et Novotny (2014). La méthode revient à utiliser un scaling permettant de passer à la limite dans les équations de Navier-Stokes compressibles et d'identifier deux régimes limites possibles suivant la valeur du nombre de Froude, en introduisant une inégalité d'entropie relative.

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.