On the stability of some groups of formal diffeomorphisms by the Birkhoff decomposition

Let G∞ be the group of one parameter identity-tangent diffeomorphisms on the line whose coefficients are formal Laurent series in the parameter ε 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∞ and give a formula for this decomposition.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 theory, this diffeomorphism is in G∞ and its Birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants.