Renormalization of Poincar E Normal Forms

In Poincar e Normal Form theory, one considers a series of transformations generated by homogeneous polynomials obtained as solution of the homological equation; such solutions are unique up to terms in the kernel of the homological operator. By a careful exploitation of higher order eeects { which requires the solution of \higher order homological equations" { we show that the Poincar e normalization procedure can be recursively iterated to produce a \renormalized" normal form; this simpliication is closely related to the non-uniqueness of the standard normal form. The proposed procedure is completely constructive. It is also shown that the discussion immediately extends on the one side to the Hamiltonian case and Birkhoo normal forms, and to the other to the equivariant setting.