arXiv subjects

Emmanuel Paul

Publications and source records attributed to Emmanuel Paul.

5 recordsLinked to original sources

arXiv2023

Dynamics of the fifth Painlev{\'e} foliation

The leaves of the Painlev{\'e} foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections. Therefore they are the fibers of the Riemann-Hilbert correspondence that sends each connection on its monodromy data, and this correspondence induces a conjugation between the dynamics of the foliation and a dynamic on a space of representations of some fundamental groupoid (a character variety). This one can be identified to a family of cubic surfaces through trace coordinates. We describe here the dynamics on the character variety related to the Painlev{\'e} V equation. We have here to consider irregular connections, and the representations of wild groupoids. We describe and compare all the dynamics which appear on this wild character variety: the tame dynamics, the confluent dynamics, the canonical symplectic dynamics and the wild dynamics.

math.DS↗

arXiv2015

Dynamics on Wild Character Varieties

In the present paper, we will first present briefly a general research program about the study of the "natural dynamics" on character varieties and wild character varieties. Afterwards, we will illustrate this program in the context of the Painlevé differential equations $P_{\rm VI}$ and $P_{\rm V}$.

math.CA↗

arXiv2009

Normal forms of foliations and curves defined by a function with a generic tangent cone

We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in each class. Then we study on these moduli space the distribution C induced by the following equivalence relation: two points are equivalent if and only if the corresponding foliations have the same analytic invariant curves up to analytical conjugacy. Therefore, the space of leaves of C is the moduli space of curves. We prove that C is rationally integrable. These rational integrals give a complete system of invariants for these generic plane curves, which extend the well-known cross-ratios between branches.

math.CV↗

arXiv2006

The Galoisian envelope of a germ of foliation: the quasi-homogeneous case

We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analytic invariants, whereas the algorithmic ones make use of formal normal forms.

math.DS↗

arXiv2003

Forme normale formelle d'une perturbation à séparatrices fixées d'un champ hamiltonien quasi-homogène

On classifie à changement de variable formel près les champs de vecteurs --ainsi que les feuilletages qu'ils définissent-- qui sont des perturbations d'un champ hamiltonien quasi-homogène $X_0$ par des termes de degré de quasi-homogénéité sup'erieur. Le degré $δ_0$ du champ $X_0$ est quelconque, mais on demande que le champ perturbé laisse encore invariant les séparatrices de $X_0$. Les formes normales présentées ici généralisent celles obtenues lorsque $X_0$ possède une partie linéaire semi-simple ou nilpotente. Nous les interpr'etons g'eom'etriquement en terme de cohomologie relative au champ initial.

math.CA↗