Non existence of convergent normal form for general germs of unipotent diffeomorphisms
germ of diffeomorphism | normal form | formal classification
A germ of diffeomorphism has convergent normal form if it is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms for every dimension greater than one such that they do not have a convergent normal form. The examples are contained in a family in which the absence of convergence normal form is linked to a geometrical phenomenon. The proof is based on several reductions; it relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms via potential theory.