Rigidity of strong and weak foliations

We consider a perturbation $ f $ of a hyperbolic toral automorphism $ L $. We study rigidity related to exceptional properties of the strong and weak stable foliations for $ f $. If the strong foliation is mapped to the linear one by the conjugacy $ h $ between $ f $ and $ L $, we obtain smoothness of $ h $ along the weak foliation and regularity of the joint foliation of the strong and unstable foliations. We also establish a similar global result. If the weak foliation is sufficiently regular, we obtain smoothness of the conjugacy along the strong foliation and regularity of the joint foliation of the weak and unstable foliations. If both conditions hold, then we get smoothness of $ h $ along the stable foliation. We also deduce a rigidity result for the symplectic case. The main theorems are obtained in a unified way using our new result on the relation between holonomies and normal forms.