TITLE:
One dimensional invariant manifolds of Gevrey type in real-analytic maps

AUTHORS:
Inma Baldomá^(1) and Ālex Haro^(2)

(1) Departament de Matemātica Aplicada I
    Universitat Politčcnica de Catalunya
    Diagonal 647
    08028 Barcelona, Spain 

(2) Departament de Matemātica Aplicada i Anālisi
    Universitat de Barcelona
    Gran Via 585
    08007 Barcelona, Spain 

E-mails: immaculada.baldoma@upc.edu, alex@maia.ub.es

ABSTRACT:
In this paper we study the basic questions of existence, uniqueness,
differentiability, analyticity and computability of the one
dimensional center manifold of a parabolic-hyperbolic fixed point of
a real-analytic map. We use the parameterization method, 
reducing the dynamics on the center manifold to a polynomial.
We prove that the asymptotic expansions of the center manifold are  
of Gevrey type. Moreover, under suitable hypothesis, we also 
prove that the asymptotic expansions correspond to a real-analytic
parameterization of an invariant curve that goes to the fixed point.
The parameterization is Gevrey at the fixed point, hence $C^\infty$.