Any (at most countable) family of pairwise transverse (even singular, with saddle or prong singularities) foliations on the plane R² admits a compactification (as the disc D²) by a circle at infinity so that every ray in a leaf tends to a point on the circle, and this compactification is unique up to two natural requirements. Thus every leaf corresponds to a pair of points on the circle. With Th. Barthelmé and K. Mann, we consider the reverse problem and we give a complete answer to the two following questions:

Q1: (realization) Under which hypotheses two sets L⁺, L⁻ of pairs of points on the circle are precisely the pairs of endpoints of leaves of two transverse foliations (we give the answer for singular, and also for nonsingular foliations, and we prove that the foliations are uniquely determined).

More important is the second question:
Q2: (completion) Under which hypotheses two sets l⁺,l⁻ or pair of points on the circle correspond the pairs of endpoints of a dense subset of leaves of two transverse foliations (singular or nonsingular, and uniqueness). The uniqueness implies that any group action on the circle preserving l⁺,l⁻ extends in an action on the disc preserving the corresponding foliations. This allows us to prove that if an action G -> Homeo_+(S¹) of a group on the circle is induced by an Anosov-like action G -> Homeo_+(D²), then this action is unique and completely determined by the action on the circle. With Th. Marty, we consider the case of 1 (singular or not) foliation and we give a complete answer to the following questions

Q3: (realization) Under which hypotheses a set L of pairs of points on the circle is precisely the set of pairs of endpoints of leaves of a foliation,

Q4: (completion) Under which hypotheses a set l of pairs of points on the circle corresponds to the pairs of endpoints of a dense subset of leaves of a foliation and we prove again the uniqueness.[-]

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.[-]

A theorem of Barbot (building on work of Ghys, Haefliger and others) says that Anosov flows on 3-manifolds are classified up to orbit equivalence by the data of a pair of transverse foliations of the plane and an action of the fundamental group of the 3-manifold. In recent work with T. Barthelmé, as well as C. Bonatti, S. Fenley and S. Frankel, we have been developing an abstract theory of Anosov-like group actions of bifoliated planes, applicable both to the study of flows and as an interesting class of foliation-preserving dynamical systems in its own right. This minicourse will explain some of this theory and the connections between flows and group actions in dimensions 1, 2 and 3. [-]

In the late '90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by both positive and negative contact structures. Additionally, if the foliation is taut then its contact approximations are tight. In this talk, I will present a converse result on constructing taut foliations from suitable pairs of contact structures. While taut foliations are rather rigid objects, this viewpoint reveals some degree of flexibility and offers a new perspective on the L-space conjecture. A key ingredient is a generalization of a result of Burago and Ivanov on the construction of branching foliations tangent to continuous plane fields, which might be of independent interest.[-]

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).[-]