[Paper Review] Invariant manifolds for analytic dynamical systems over ultrametric fields

Helge Glöckner|ArXiv.org|Aug 20, 2008

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TL;DR

This paper establishes the existence and structure of invariant manifolds—specifically a-stable, centre-stable, and centre manifolds—for analytic dynamical systems over complete ultrametric fields. Using non-archimedean analysis and spectral conditions on the tangent map, it proves that under a-hyperbolicity, the a-stable set is an analytic submanifold tangent to the a-stable subspace, generalizing classical stable manifold theorems to the ultrametric setting with applications to Lie groups over local 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).

Motivation & Objective

• To extend the classical stable manifold theory to analytic dynamical systems over complete ultrametric fields, particularly in positive characteristic.
• To define and construct a-stable, centre-stable, and centre manifolds in the non-archimedean setting using spectral and geometric conditions on the tangent map.
• To provide a foundation for partially hyperbolic dynamical systems over ultrametric fields by generalizing hyperbolicity concepts to a-hyperbolicity.
• To prove that the a-stable set is an analytic submanifold tangent to the a-stable subspace under a-hyperbolicity of the linearized map at a fixed point.
• To establish that local and global stable manifolds are invariant under the dynamics and admit a recursive structure via inverse iteration.

Proposed method

• Uses a-hyperbolicity of the tangent map $T_p(f)$, defined via decomposition of the tangent space into $a$-stable and $a$-unstable subspaces with norm conditions on the operator norms.
• Applies non-archimedean analysis and ultrametric norms satisfying the strong triangle inequality to define the dynamics in a Banach space model.
• Employs the contraction mapping principle in sequence spaces to construct the $a$-stable manifold as a graph over the $a$-unstable subspace.
• Uses the inverse function theorem and analytic implicit function theory to show that the invariant manifold is an immersed analytic submanifold.
• Applies spectral theory to relate $a$-hyperbolicity to the spectrum of the linear map in finite dimensions, ensuring $a \neq |\lambda|$ for all eigenvalues $\lambda$.
• Constructs the manifold via backward iteration: $W_a^s = \bigcup_{n=0}^\infty f^{-n}(\Omega)$ for a suitable neighborhood $\Omega$ of $p$.

Experimental results

Research questions

• RQ1 Under what conditions does an analytic diffeomorphism on a manifold over an ultrametric field admit an invariant $a$-stable manifold?
• RQ2 How can the classical notion of hyperbolicity be adapted to the ultrametric setting to ensure the existence of stable manifolds?
• RQ3 What is the precise structure of the $a$-stable set $W_a^s(f,p)$, and when is it an analytic submanifold?
• RQ4 How does the choice of $a \in (0,1]$ affect the geometry and dynamics of the invariant manifold?
• RQ5 Can the theory be extended to centre-stable and centre manifolds in the presence of a non-hyperbolic part?

Key findings

• The $a$-stable set $W_a^s(f,p)$ is an immersed analytic submanifold of $M$ tangent to the $a$-stable subspace $T_p(M)_{a,s}$ of the tangent space.
• The manifold structure on $W_a^s(f,p)$ is unique and invariant under $f$, with $f|_{W_a^s}$ being an analytic diffeomorphism onto itself.
• Each neighborhood of $p$ in $W_a^s$ contains a submanifold $\Omega$ such that $W_a^s = \bigcup_{n=0}^\infty f^{-n}(\Omega)$, reflecting the backward dynamics.
• For $a=1$, the $1$-stable manifold $W^s$ coincides with the classical stable manifold, and $T_p(f)$ is hyperbolic if and only if $1 \notin \{|\lambda| : \lambda \text{ eigenvalue of } T_p(f) \otimes_{\mathbb{K}} \mathrm{id}_{\overline{{\mathbb{K}}}}\}$.
• In the finite-dimensional case, $a$-hyperbolicity is equivalent to $a \neq |\lambda|$ for all eigenvalues $\lambda$ of the linearized map in an algebraic closure.
• The $a$-unstable manifold is constructed as the graph of an analytic function over the unstable subspace, using backward iteration and the contraction principle.