Diagonalizable Flows on Locally Homogeneous Spaces and Number
Diagonalizable flows on locally homogeneous spaces and number theory Manfred Einsiedler and Elon Lindenstrauss∗ Abstract. We discuss dynamical properties of actions of diagonalizable groups on locally ho- mogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications. Mathematics Subject Classification (2000). 37D40, 37A45, 11J13, 81Q50. Keywords. Invariant measures, locally homogeneous spaces, Littlewood’s conjecture, quantum unique ergodicity, distribution of periodic orbits, ideal classes, entropy. 1. Introduction Flows on locally homogeneous spaces are a special kind of dynamical systems. The ergodic theory and dynamics of these flows are very rich and interesting, and their study has a long and distinguished history. What is more, this study has found nu- merous applications throughout mathematics. The spaces we consider are of the form \G where G is a locally compact group and a discrete subgroup of G. Typically one takes G to be either a Lie group, a linear algebraic group over a local field, or a product of such. Any subgroup H<Gacts on \G and this action is precisely the type of action we will consider here. One of the most important examples which features in numerous number theoretical applications is the space PGL(n, Z)\ PGL(n, R) which can be identified with the space of lattices in Rn up to homothety. Part of the beauty of the subject is that the study of very concrete actions can have meaningful implications. For example, in the late 1980s G. A. Margulis proved the long-standing Oppenheim conjecture by classifying the closed orbits of the group of matrices preserving an indefinite quadratic form in three variables in PGL(3, Z)\ PGL(3, R) – a concrete action of a three-dimensional group on an eight- dimensional space.
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