Global rigidity of smooth ${\mathbb{Z}}\ltimes _{\unicode{x3bb}}{\mathbb{R}}$-actions on ${\mathbb{T}}^{2}$

Abstract For $\unicode{x3bb}>1$ , we consider the locally free ${\mathbb Z}\ltimes _\unicode{x3bb} \mathbb R$ actions on $\mathbb T^2$ . We show that if the action is $C^r$ with $r\geq 2$ , then it is $C^{r-\epsilon }$ -conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along the 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.