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Theses

Sur le groupoide de Galois d'un feuilletage

Abstract : B. Malgrange defines the D-envelope or galoisian envelope of an analytical dynamical system. Roughly speaking, this is the algebraic hull of the dynamical system.

This thesis is in three parts.
In the first part, germs of Lie D-groupoids and D-envelopes of germs of diffeomorphisms of (C,0) are investigated. We obtain a characterisation of these envelopes on résurgent coefficients of the normalising power serie. This results provides a methode to compute rational differential equations satisfied by some special diffeomorphisms (given by Jean Écalle).

In the second, the D-envelope of a rational map R of the complex projective line is computed. The rational maps characterised by a finitness property of their D-envelope appear to be the integrable ones (Tchebitchev polynomials and Lattès examples).

In the third, one considere a germ of codimension one holomorphic foliation F. We study the Galois groupoid of F. In a first part, we define the transversal rank of the Galois groupoid. In a second part, we prove that the finitness of this rank is equivalent to the existence of a meromorphic Godbillon-Vey sequence of length less or equal than three. In a third part, relations between the Galois groupoid and the Kolchin theory are given.
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Submitted on : Thursday, March 23, 2006 - 12:45:11 PM
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Guy Casale. Sur le groupoide de Galois d'un feuilletage. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2004. Français. ⟨tel-00012021⟩

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