Local rigidity of certain solvable group actions on tori

TL;DR

This paper investigates the local rigidity of certain solvable group actions on tori, showing that small smooth perturbations are smoothly conjugate to the original affine actions, up to a proportional time change.

Contribution

It establishes a weak form of local rigidity for affine actions generated by irreducible toral automorphisms and linear flows on tori.

Findings

Smooth perturbations are conjugate to affine actions.

Rigidity holds up to proportional time change.

Results apply to actions generated by specific solvable groups.

Abstract

In this paper, we study a local rigidity property of $\mathbb Z \ltimes_\lambda \mathbb R$ affine action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace. Such an action exhibits a weak version of local rigidity, i.e., any smooth perturbations close enough to an affine action is smoothly conjugate to the affine action up to proportional time change.