Invariant manifolds for analytic dynamical systems over ultrametric fields

Abstract

We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an ultrametric Banach space over K, an analytic self-map f of M, and a fixed point p of f. Under suitable conditions on the tangent map of f at p, we construct a centre-stable manifold, a centre manifold, respectively, an r-stable manifold around p, for a given positive real number r not exceeding 1. The invariant manifolds are useful in the theory of Lie groups over local fields, where they allow results to be extended to the case of positive characteristic which previously were only available in characteristic zero (i.e., for p-adic Lie groups).