NORMAL FORMS OF ANALYTIC PERTURBATIONSOF QUASIHOMOGENEOUS VECTOR FIELDS:RIGIDITY, INVARIANT ANALYTIC SETS ANDEXPONENTIALLY SMALL APPROXIMATION

摘要

In this article, we study germs of holomorphic vector fields which are "higher order" perturbations of a quasihomogeneous vector field in a neighborhood of the origin of C~n, fixed point of the vector fields. We define a "Diophantine condition" on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. We study the normal form problem relatively to S. We give a condition on S that ensures that there always exists an holomorphic transformation to a normal form. If this condition is not satisfied, we also show, that under some reasonable assumptions, each perturbation of S admits a Gevrey formal normalizing transformation to a Gevrey formal normal form. Finally, we give an exponentially good approximation of the dynamic by a partial normal form.