Invariant manifolds for finite-dimensional non-archimedean dynamical systems

TL;DR

This paper studies invariant manifolds such as stable and centre manifolds for finite-dimensional analytic dynamical systems over non-archimedean fields, with applications to Lie groups over local fields.

Contribution

It provides new results on invariant manifolds in finite-dimensional non-archimedean dynamical systems, emphasizing the ultrametric setting.

Findings

Existence of invariant manifolds in finite-dimensional ultrametric manifolds.

Characterization of stable, centre-stable, and centre manifolds in this setting.

Applications to Lie groups over totally disconnected local fields.

Abstract

Let M be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field. Let f be an analytic diffeomorphism from M onto itself and p be a fixed point of f. We discuss invariant manifolds around p, like stable manifolds, centre-stable manifolds and centre manifolds, with an emphasis on results specific to the case that M has finite dimension. The results have applications in the theory of Lie groups over totally disconnected local fields.