On Measures Invariant under Diagonalizable Actions – the Rank One Case and the General Low Entropy Method

We consider measures on locally homogeneous spaces Γ\G which are invariant and have positive entropy with respect to the action of a single diagonalizable element a ∈ G by translations, and prove a rigidity statement regarding a certain type of measurable factors of this action. This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [Lin2, EKL] is used to classify positive entropy measures invariant under a one parameter group with an additional recurrence condition for G = G1 × G2 with G1 a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.