Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians

TL;DR

This paper extends the theory of normal forms and stability analysis from Gevrey to finitely differentiable Hamiltonians, providing polynomial bounds on stability time and invariant manifold splitting.

Contribution

It introduces a new method to derive normal forms with polynomially small remainders for finitely differentiable Hamiltonians, advancing stability and splitting estimates.

Findings

Polynomial bounds on stability time for action variables

Polynomial bounds on splitting of invariant manifolds

Extension of normal form techniques to finitely differentiable case

Abstract

This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at a quasi-periodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.