We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii’s work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno’s condition (A). In dimension 2, no condition is required since, according to Stróżyna–Żołladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations.

Let \({X = \lambda_{1} x_{1}{\frac{\partial}{\partial {x_1}}} + \lambda_2 x_2 \frac{\partial}{\partial {x_2}} + O(|x|^2)}\) be an analytic vector field near x = 0. We suppose that the linear part of this vector field has real eigenvalues λ1, λ2 and that the ratio \({\eta = -\frac{\lambda_1}{\lambda_2}}\) is a positive irrational number. In a previous paper of the first author and P. De Maesschalck, it was shown that any analytic saddle can be conjugated analytically to a form ‘as close as desired’ to the formal normal form. In this paper we will iterate and renormalize these conjugacies. The iteration of this procedure will be strongly connected to the diophantine properties of η and we will establish the convergence of this process. A consequence of this convergence will be the two dimensional version of the by now classical linearization theorem of Bruno.

We explore the convergence/divergence of the normal form for a singularity of a vector field on $\C^n$ with nilpotent linear part. We show that a Gevrey-$\alpha$ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey-$1+\alpha$ type with the use of a Gevrey-$1+\alpha$ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegel's theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1 : -1 resonant saddle. (C) 2010 Elsevier Inc. All rights reserved.

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.