A KAM Theorem for finitely differentiable Hamiltonian systems

TL;DR

This paper extends KAM theory to finitely differentiable Hamiltonian systems, proving the persistence of a measure of invariant tori under small perturbations, generalizing previous results that only guaranteed a single torus.

Contribution

It establishes a measure estimate for invariant tori in finitely differentiable systems, broadening the scope of KAM theory beyond analytic cases.

Findings

Persistence of a Cantor family of KAM tori with positive measure

Extension of Salamon's result from a single torus to a family

Measure of invariant tori scales with perturbation size as O(ε^{1/2 - ν/l})

Abstract

Given $l>2\nu>2d\geq 4$, we prove the persistence of a Cantor--family of KAM tori of measure $O(\varepsilon^{1/2-\nu/l})$ for any non--degenerate nearly integrable Hamiltonian system of class $C^l(\mathscr D\times\mathbb{T}^d)$, where $\mathscr D\subset \mathbb{R}^d$ is a bounded domain, provided that the size $\varepsilon$ of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.