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

For $\lambda>1$, we consider the locally free ${\mathbb Z}\ltimes_\lambda{\mathbb R}$ actions on ${\mathbb T}^2$. We show that, if the action is $C^r$ with $r\geq2$, 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.