Invariant Measures for Actions of Higher Rank Abelian Groups

The first part of the paper begins with an introduction into Anosov actions of zk and JRk and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed proof of a modified version of the main theorem from [KS3] for actions by toral automorphisms of with applications to rigidity of the measurable structure of such actions with respect to Lebesgue measure. In the second part principal technical tools for studying non uniformly hyperbolic actions of zk and JRk are introduced and developed. These include Lyapunov characteristic exponents, nonstationary normal forms and Lyapunov Hoelder structures. At the end new rigidity results for 2 2 actions on three-di\flensional manifolds are outlined.