KAM Theorem for Gevrey Hamiltonians

TL;DR

This paper proves a KAM theorem for Gevrey Hamiltonians, establishing the existence of invariant tori with Diophantine frequencies and a Gevrey normal form, leading to stability results in near-integrable systems.

Contribution

It extends KAM theory to Gevrey regular Hamiltonians, providing a Gevrey smooth invariant tori family and a symplectic normal form near these tori.

Findings

Existence of Gevrey smooth invariant tori with Diophantine frequencies.

Construction of a Gevrey normal form near the invariant tori.

Effective stability of quasiperiodic motion in the neighborhood.

Abstract

We consider Gevrey perturbations $H$ of a completely integrable Gevrey Hamiltonian $H_0$. Given a Cantor set $\Omega_\kappa$ defined by a Diophantine condition, we find a family of KAM invariant tori of $H$ with frequencies $\omega\in \Omega_\kappa$ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda$ of the invariant tori. This leads to effective stability of the quasiperiodic motion near $\Lambda$.