An element g of an abstract group G is a distortion element if there exists a finite family S in G such that g belongs to the subgroup generated by S and the wordlength of gn (w.r.t. S) grows sublinearly in n. In this talk, we will be interested in the distortion elements of the group of Cr orientation-preserving diffeomorphisms of the closed interval, for different values of r. In particular, we will present some natural obstructions to distortion (such that the presence of hyperbolic fixed points in C1 regularity and the positivity of the so-called asymptotic distortion in C2 regularity (and higher)), and we will wonder whether they are the only ones.[-]

Studying the (closure of the) (semi-)conjugacy class of a given group action on a 1-manifold is interesting from many points of view. Depending on the manifold and/or the differentiability involved, one is faced with problems concerning small denominators, growth of groups / orbits, distortion elements, bounded cohomology, group orderability, etc. In this minicourse we will explore several general results on this topic such as the $C^1$ smoothing via (semi-)conjugacies of small group actions and obstructions in class $C^2$ and higher. We will also explore some of the ideas involved in the proof of the connectedness of the space of $\mathbb{Z}^d$ actions by diffeomorphisms of $C^{1+ac}$ regularity (obtained in collaboration with H. Eynard-Bontemps).[-]

Studying the (closure of the) (semi-)conjugacy class of a given group action on a 1-manifold is interesting from many points of view. Depending on the manifold and/or the differentiability involved, one is faced with problems concerning small denominators, growth of groups / orbits, distortion elements, bounded cohomology, group orderability, etc. In this minicourse we will explore several general results on this topic such as the $C^1$ smoothing via (semi-)conjugacies of small group actions and obstructions in class $C^2$ and higher. We will also explore some of the ideas involved in the proof of the connectedness of the space of $\mathbb{Z}^d$ actions by diffeomorphisms of $C^{1+ac}$ regularity (obtained in collaboration with H. Eynard-Bontemps).[-]

Studying the (closure of the) (semi-)conjugacy class of a given group action on a 1-manifold is interesting from many points of view. Depending on the manifold and/or the differentiability involved, one is faced with problems concerning small denominators, growth of groups / orbits, distortion elements, bounded cohomology, group orderability, etc. In this minicourse we will explore several general results on this topic such as the $C^1$ smoothing via (semi-)conjugacies of small group actions and obstructions in class $C^2$ and higher. We will also explore some of the ideas involved in the proof of the connectedness of the space of $\mathbb{Z}^d$ actions by diffeomorphisms of $C^{1+ac}$ regularity (obtained in collaboration with H. Eynard-Bontemps).[-]

We consider infinite interval exchange transformations (IETs) obtained as a composition of a finite IET and the von Neumann-Kakutani map, called rotated odometers, and study their dynamical and ergodic properties by means of an associated Bratteli-Vershik system. We show that every rotated odometer is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends, with respect to the Lebesgue measure. This is one motivation for the study of rotated odometers. We also prove a few results about the factors of the unique minimal subsystem of a rotated odometer. This is joint work with Henk Bruin.[-]

The study of the path-connectedness of the space of $C^{r}$ actions of $\mathbb{Z}^{2}$ on the interval [0,1] plays an important role in the classification of codimension 1 foliations on 3 manifolds. One way to deform actions is by conjugation. If an action can be brought arbitrarily close to the trivial one by conjugation, it is said to be quasi-reducible. In this talk, we will present and compare obstructions to quasi-reducibility in different regularity classes, and draw conclusions concerning the initial connectedness problem.[-]

The applications of renormalization ideas in Dynamical Systems became increasingly popular after 1979, and, since then, they played an important role in the study of several classes of low-dimensional systems.Very roughly speaking, the philosophy of renormalization is that, after appropriate rescalings, the long time behaviors at short scales of certain systems are dictated by other systems within a fixed class S of systems. In particular, such a renormalization procedure can iterated and, as it turns out, the phrase portraits of those systems whose successive renormalizations tend to stay in a compact portion of S can often be reasonably described (”plough in the dynamical plane to harvest in the parameter space”, A. Douady).In this minicourse, we shall illustrate these ideas by explaining the com-mon strategy of ”recurrence of renormalization to compact sets” behind two different results:
1.the solutions of Masur and Veech in 1982 to Keane's conjecture of unique ergodicity of almost all interval exchange transformations;
2. the solution of Moreira–Yoccoz in 2001 to Palis' conjecture on the prevalence of stable intersections of pairs of dynamical Cantor sets whose Hausdorff dimensions are large.[-]

​An almost Anosov flow is a flow having continuous flow-invariant splitting of the tangent bundle with exponential expansion/contraction in the unstable/stable direction, except for a finite number (in our case a single) periodic orbits. Roughly, almost Anosov flows are perturbed Anosov flows, where the perturbation is local around these periodic orbits, making them neutral. For this type of flows, we obtain limit theorems (stable, standard and non-standard CLT) for a large class of (unbounded) observables. I will present these results stressing on the method of proof. This is joint work with H. Bruin and M. Todd.[-]