Grobman-Hartman theorems for diffeomorphisms of Banach spaces over valued fields

Abstract

Consider a local diffeomorphism f of an ultrametric Banach space over an ultrametric field, around a hyperbolic fixed point x. We show that, locally, the system is topologically conjugate to the linearized system. An analogous result is obtained for local diffeomorphisms of real p-Banach spaces (like l^p) for 0 < p =< 1. More generally, we obtain a local linearization if f is merely a local homeomorphism which is strictly differentialble at a hyperbolic fixed point x. Also a new global version of the Grobman-Hartman theorem is provided. It applies to Lipschitz perturbations of hyperbolic automorphisms of Banach spaces over valued fields. The local conjugacies H constructed are not only homeomorphisms, but H and H^{-1} are Hoelder. We also study the dependence of H and H^{-1} on f (keeping x and f'(x) fixed).