In a recent work with B. Kalinin and A. Katok we showed that:
Theorem: an ergodic invariant measure by a $\Z^k$ action on a $k+1$ dimensional manifold $M$, $k\geq 2$ with positive entropy and Lyapunov exponents in general position is absolutely continuous with respect to Lesbesgue measure.
The only examples of actions of this type that are known are linear actions on tori and some constructions starting with such linear actions blowing up some finite orbits and following with some gluing procedure or taking some finite quotient.
In a joint work with A. Katok we are able to prove that from a measure theoretic point of view these are the only possible examples. We prove that if an action is as in the Theorem then up to taking finite index subgroup and restricting to some period the action is measurably conjugated to a linear action on an infratorus (i.e. a finite algebraic quotient of a torus). Moreover we can show the measurable conjugacy extends to a continuous map from an open invariant subset $U$ of the manifold $M$ to some open invariant subset $V$ of the torus which is necessarily the complement of a finite set. Moreover $U$ is homeomorphic to $V$.
We get several corollaries from this statement, for instance there are no such actions on spheres, if such an action is Anosov then it is smoothly conjugated to actions by linear automorphisms of the torus.
During the talk we shall describe the examples and we will try to show the ideas of the proof of this result and discuss the corollaries