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

Abstract

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.