Local differentiable rigidity for most algebraic hyperbolic actions of Z^k and R^k was proved in [KS]. In this proof topological conjugacy transversal to the orbit foliation is established first and then it is shown to be smooth using the nonstationary normal forms theory. This method is not sufficient for partially hyperbolic actions whose elements are far from being structurally stable. We introduce a new method where first rigidity of central foliations is established following hyperbolic methods and then perturbations along the central direction are tackled by a KAM type iterative scheme with a version of cocycle rigidity used to solve the linearized equation. We will also discuss a possibility of doing the whole proof of rigidity based on an iterative scheme without an a priori existence of a topological conjugacy.
This is work in progress, partly in collaboration with the Ph.D. student Danijela Damjanovic.
[KS] A. Katok and R.J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions , Proc. Steklov Inst. Math. (1997), 287--314.