Linearization of germs of hyperbolic vector fields

We develop a normal form to express asymptotically a conjugacy between a germ of resonant vector field and its linear part. We show that such an asymptotic expression can be written in terms of functions of the Logarithmic Mourtada type. To cite this article: P Bonckaert et al., C. R. Acad. Sci. Paris, Ser. I336 (2003). (C) 2003 Academie des sciences/Editions scientifiques et medicales Elsevier SAS. All rights reserved.MathematicsSCI(E)0ARTICLE119-2233

Linearization of hyperbolic resonant fixed points of diffeomorphisms with related Gevrey estimates in the planar case

We show that any germ of smooth hyperbolic diffeomophism at a fixed point is conjugate to its linear part, using a transformation with a Mourtada type functions, which (roughly) means that it may contain terms like $x \log |x|$. Such a conjugacy admits a Mourtada type expansion. In the planar case, when the fixed point is a p:-q resonant saddle, and if we assume that the diffeomorphism is of Gevrey class, we give an upper bound on the Gevrey estimates for this expansion

On the index of finite determinacy of vector fields with resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R-n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable