Measurable rigidity and disjointness for ℤ^k actions by toral automorphisms

We investigate joinings of strongly irreducible totally non-symplectic Anosov ℤ^k, k ≥ 2 actions by toral automorphisms. We show that the existence of a non-trivial joining has strong implications for these actions, in particular, that the restrictions of the actions to a finite index subgroup of ℤ^k are conjugate over Q. We also obtain a description of the joining measures modulo the classification of zero entropy measures for the actions.