Local Rigidity for Simultaneous Diophantine Translations on Tori of Arbitrary Dimension

TL;DR

This paper proves that small smooth perturbations of certain Diophantine translation actions on tori are smoothly conjugate to the original actions, extending previous results to higher dimensions and ranks.

Contribution

It generalizes existing rigidity results for toral actions to higher-dimensional tori and higher rank abelian groups under Diophantine conditions.

Findings

Perturbations are smoothly conjugate to original actions under natural conditions.

Extends Moser's and Karaliolios's results to higher dimensions and ranks.

Provides a framework for understanding rigidity in higher-dimensional toral dynamics.

Abstract

We show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action on the torus by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets. This generalizes recent result of Karaliolios [4] of the action generators to higher rank abelian actions, and the result of Moser [8] to higher dimensional tori.