Local Rigidity of Diophantine translations in higher dimensional tori

TL;DR

This paper proves that small smooth perturbations of Diophantine rotations on higher-dimensional tori are smoothly conjugate to the original rotation, using a KAM scheme ensuring stability of the rotation vector.

Contribution

It establishes a smooth conjugacy result for perturbed Diophantine rotations on higher-dimensional tori, extending rigidity results in dynamical systems.

Findings

Perturbations preserving the rotation vector are smoothly conjugate to the original rotation.

The KAM scheme converges in the $C^{ abla}$ topology, ensuring smooth linearization.

Invariant measures with the same rotation vector facilitate the linearization process.

Abstract

We prove a theorem asserting that, given a Diophantine rotation $\alpha $ in a torus $\T ^{d} \equiv \R ^{d} / \Z ^{d}$, any perturbation, small enough in the $C^{\infty}$ topology, that does not destroy all orbits with rotation vector $\alpha$ is actually smoothly conjugate to the rigid rotation. The proof relies on a K.A.M. scheme (named after Kolmogorov-Arnol'd-Moser), where at each step the existence of an invariant measure with rotation vector $\alpha$ assures that we can linearize the equations around the same rotation $\alpha$. The proof of the convergence of the scheme is carried out in the $C^{\infty}$ category.