Global rigidity of smooth $\mathbb{Z}{⋉}_{\lambda }\mathbb{R}$-actions on ${\mathbb{T}}^2$

Abstract

For $\lambda >1$, we consider the locally free $\mathbb{Z}{⋉}_{\lambda }\mathbb{R}$ actions on ${\mathbb{T}}^2$. We show that, if the action is $C^r$ with $r\ge 2$, then it is $C^{r-\epsilon }$-conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along expanding eigen-direction of the automorphism. In contrast, there exists a $C^{1+\alpha }$-action which is semi-conjugate, but not topologically conjugate to an affine action.