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

TL;DR

This paper extends the Grobman-Hartman theorem to ultrametric and p-Banach spaces, establishing local and global topological conjugacy between nonlinear and linear systems near hyperbolic fixed points, with Hölder continuity.

Contribution

It generalizes the classical theorem to ultrametric Banach spaces and p-Banach spaces, providing new local and global linearization results with Hölder conjugacies.

Findings

Local conjugacy to linear systems in ultrametric Banach spaces

Global conjugacy for Lipschitz perturbations of hyperbolic automorphisms

Hölder continuity of the conjugacy and its inverse

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).