Global rigidity of smooth ${ Z}_λ{ R}$-actions on ${ T}^2$

For λ>1, we consider the locally free Z_λ R actions on T². We show that, if the action is Cʳ with r≥2, then it is Cr-ε-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 C1+α-action which is semi-conjugate, but not topologically conjugate to an affine action.