On the stability of some groups of formal

diffeomorphisms by the Birkhoff decomposition

Let $G_{\infty}$ be the group of one parameter formal identity-tangent diffeomorphisms on the line whose coefficients are formal Laurent series in the parameter $\varepsilon$ with a pole of finite order at $0$. It is well-known that the Birkhoff decomposition can be defined in such a group. We investigate the stability of the Birkhoff decomposition in subgroups of $G_{\infty}$ and give a formula for this decomposition. To do so, we will use Jean Ecalle's formalism of Mould expansions and Tree expansions These results are strongly related to renormalization in quantum field theory, since it was proved by A. Connes and D. Kreimer that, after dimensional regularization, the unrenormalized effective coupling constants are the image by a formal identity-tangent diffeomorphism of the coupling constants of the theory. In the massless $\phi^3_6$ theory, this diffeomorphism is in $G_{\infty} $ and its Birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants.

Retour page: Séminaire Mulhousien de Mathématiques

Dernières modifications / Last modifications : 04 Octobre 2007