LS category, foliated spaces and transverse invariant measure

The LS category is a homotopy invariant of topological spaces introduced by Lusternik and Schnirelmann in 1934, which was originally motivated by problems of variational calculus. It is defined as the minimum number of contractible open subsets needed to cover a space. Besides its original variational application, it became an important tool in homotopy theory, and it was applied in other di↵erent areas like robotics. Many variants of the LS category has been given; in particular, E. Mac´ıas and H. Colman introduced a tangential version for foliations, where they used leafwise contractions to transversals. In this thesis, the following new versions of the tangential LS category are introduced

Every noncompact surface is a leaf of a minimal foliation

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of open surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely $C^2$-smoothable.Comment: 40 pages, 8 figures. Overall redaction improved in this version. The topology of the ambient manifold is treated with more care and we show examples with nontrivial Euler clas

Groups with infinitely many ends acting analytically on the circle

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.Comment: We can now make a precise reference to Deroin's work arXiv:1811.10298. 54 pages, 2 figure

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group $\mathsf{Diff}^\omega_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^\omega_+(\mathbb S^1)$ acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work.Comment: v3 36 pages, 5 figures; cosmetic change