Formal classification of unipotent parameterized diffeomorphisms

Abstract

We provide a complete system of invariants for the formal classification of complex analytic unipotent germs of diffeomorphism at $\cn{n}$ fixing the orbits of a regular vector field. We reduce the formal classification problem to solve a linear differential equation. Then we describe the formal invariants; their nature depends on the position of the fixed points set $Fix \phi$ with respect to the regular vector field preserved by $\phi$. We get invariants specifically attached to higher dimension ($n \geq 3$) although generically they are analogous to the one-dimensional ones.