# Positive measure of KAM tori for finitely differentiable Hamiltonians

## TL;DR

This paper improves the understanding of the persistence of KAM tori in finitely differentiable Hamiltonian systems, showing that a positive measure set of such tori persists under less restrictive smoothness conditions than previously known.

## Contribution

It demonstrates that the persistence of positive measure of KAM tori requires only the perturbation to be $C^l$ and the integrable part to be $C^{l+2}$, relaxing earlier smoothness assumptions.

## Key findings

- Positive measure of KAM tori persists under $C^l$ perturbations.
- Relaxed smoothness conditions compared to prior results.
- Extension of KAM theory to finitely differentiable Hamiltonians.

## Abstract

Consider an integer $n \geq 2$ and real numbers $\tau>n-1$ and $l>2(\tau+1)$. Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^l$. Under the stronger assumption that the system is a $C^{l+\tau}$ perturbation of an analytic integrable system, P{\"o}schel proved the persistence of a set of positive measure of Diophantine tori. We improve the last result by showing it is sufficient for the perturbation to be of class $C^{l}$ and the integrable part to be of class $C^{l+2}$.