Generalized Flow-Box property for singular foliations

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety $X$ (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (concerning the strengthening of a Theorem of Mattei) about the existence of local slices for a (non-compact) Lie group action.Comment: Changes in Section

Regularization of Discontinuous Foliations: Blowing up and Sliding Conditions via Fenichel Theory

We study the regularization of an oriented 1-foliation $\mathcal{F}$ on $M \setminus \Sigma$ where $M$ is a smooth manifold and $\Sigma \subset M$ is a closed subset, which can be interpreted as the discontinuity locus of $\mathcal{F}$. In the spirit of Filippov's work, we define a sliding and sewing dynamics on the discontinuity locus $\Sigma$ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions.Comment: 32 page

On the existence of canard solutions

We study the existence of global canard surfaces for a wide class of real singular perturbation problems. These surfaces define families of solutions which remain near the slow curve as the singular parameter goes to zero

On the existence of canard solutions

We study the existence of global canard surfaces for a wide class of real singular perturbation problems. These surfaces define families of solutions which remain near the slow curve as the singular parameter goes to zero

PSL(2, ℂ), the exponential and some new free groups

We prove a normal form result for the groupoid of germs generated by PSL(2, ) and the exponential map. We discuss three consequences of this result: (1) a generalization of a result of Cohen about the group of translations and powers, which gives a positive answer to a problem posed by Higman; (2) as proof that the subgroup of Homeo(, +¥) generated by the positive affine maps and the exponential map is isomorphic to an HNN-extension; (3) a finitary version of the immiscibility conjecture of Ecalle–Martinet–Moussu–Ramis