Normal form of holomorphic dynamical systems

Abstract

This article represents the expanded notes of my lectures at the ASI "Ham- iltonian Dynamical Systems and applications". We shall present various recent re- sults about normal forms of germs of holomorphic vector fields at a fixed point in C n . We shall explain how relevant it is for geometric as well as for dynamical pur- pose. We shall first give some examples and counter-examples about holomorphic conjugacy. Then, we shall state and prove a main result concerning the holomor- phic conjugacy of a commutative family of germs of holomorphic vector fields. For this, we shall explain the role of diophantine condition and the notion of singular complete integrability.