Diagonal Actions on Locally Homogeneous Spaces

Contents 1. Introduction 1 2. Ergodic theory: some background 4 3. Entropy of dynamical systems: some more background 6 4. Conditional Expectation and Martingale theorems 12 5. Countably generated σ-algebras and Conditional measures 14 6. Leaf-wise Measures, the construction 19 7. Leaf-wise Measures and entropy 37 8. The product structure 61 9. Invariant measures and entropy for higher rank subgroups A, the high entropy method 67 10. Invariant measures for higher rank subgroups A, the low entropy method 79 11. Combining the high and low entropy methods. 86 12. Application towards Littlewood's Conjecture 89 13. Application to Arithmetic Quantum Unique Ergodicity 94 References 104 1. Introduction 1.1. In these notes we present some aspects of work we have conducted, parts jointly with Anatole Katok, regarding dynamics of higher rank diagonalizable groups on homogeneous spaces Γ\G. A prototypical example of such an action is the action of the group of determinant one diagonal matrices A on the space of unit volume lattices in R n for n ≥ 3 which can be identified with the quotient space SL(n, Z)\ SL(n, R). More specifically, we consider the problem of classifying measures invariant under such an action, and present two of the applications of this measure classification. There have been several surveys on this topic, including some that we have written (specifically, [Lin05] and [EL06]). For this reason we will be brief in our historical discussions and the discussion of the important work of the pioneers of the subject. 1.2. Let G be a linear algebraic group R, and Γ < G a lattice (i.e. a discrete, finite covolume subgroup). One can more generally consider for any subgroup H < G, in particular for any algebraic subgroup, the action of H on the symmetric space Γ\G. Ratner's landmark measure classification theorem (which is somewhat more general as it considers the case of G a general Lie group) states the following: 1.3. Theorem (M. Ratner [Rat91]). Let G, Γ be as above, and let H < G be an algebraic subgroup generated by one parameter unipo-tent subgroups. Then any H-invariant and ergodic probability measure µ is the natural (i.e. L-invariant) probability measure on a single orbit of some closed subgroup L < G (L = G is allowed). We shall call a probability measure of the type above (i.e. supported on a single orbit of its stabilizer group) homogeneous. 1.4. For one parameter diagonalizable …