On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method

Abstract

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 = G 1 × G 2 with G 1 a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.