Grobman-Hartman theorems for diffeomorphisms of Banach spaces over valued fields. (English) Zbl 1321.37015
Shamseddine, Khodr (ed.), Advances in ultrametric analysis. Selected papers based on the presentations at the 12th international conference on \(p\)-adic functional analysis, Winnipeg, MB, Canada, July 2–6, 2012. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9142-1/pbk; 978-1-4704-1024-7/ebook). Contemporary Mathematics 596, 79-101 (2013).
Summary: 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 \(\ell^p\)) for \(p\in\left]0,1\right]\). More generally, we obtain a local linearization if \(f\) is merely a local homeomorphism which is strictly differentiable 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 Hölder. We also study the dependence of \(H\) and \(H^{-1}\) on \(f\) (keeping \(x\) and \(f'(x)\) fixed).
For the entire collection see [Zbl 1273.00050].
For the entire collection see [Zbl 1273.00050].
MSC:
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
| 26E30 | Non-Archimedean analysis |
| 46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |
| 46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
