Diagonalizable flows on locally homogeneous spaces and number theory

Abstract

Abstract.We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous 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.

Local-global principles for representations of quadratic forms

“…However, there is nevertheless a natural approach to the case n = m + 2, replacing our use of Ratner theory by the emerging theory of torus rigidity (see, in particular, the survey and announcement [9]). The idea is to replace the use of Q p by a product of two completions Q p × Q q , and consider the action of SO 2 (Q p )× SO 2 (Q q ) on Γ\SO n (Q p )× SO n (Q q ).…”

The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture

“…In order to enunciate precise theorems about our dynamical systems, we shall need to make use of the notion of positive entropy. Let us briefly give an incomplete survey of this notion; for more, see [6,Section 3] (in the context of homogeneous spaces) and [25] (as a general reference).…”

“…Several applications of this type of dynamics are surveyed in [6]. For now it is worth commenting on two rather different contexts in number theory where exactly the same dynamics arise:…”

Equidistribution, <em>L</em>-functions and ergodic theory: on some problems of Yu. Linnik

“…Much of this joint work is based on the recent work of Einsiedler and Lindenstrauss on classification of invariant measures for higher rank tori, which is discussed in their contribution to these proceedings [EL06]. An important difference between the higher rank case and the rank 1 case considered by Linnik is the phenomenon of measure rigidity; in general, the actions of higher rank tori are far more rigid (have fewer invariant measures and closed invariant sets) than the actions of rank 1 tori.…”

“…If H is a torus, the emerging theory of measure rigidity for torus actions (see in particular [EL06], [EKL06]) can often substitute for Ratner theory. This requires an extra input, positive entropy, and has two further disadvantages (compared to "Ratner theory") that might be noted:…”