# Meeting Details

Title: Equidistribution of joinings under off-diagonal polynomial flows Center for Dynamics and Geometry Seminars Tim Austin, Brown University Furstenberg proved the Multiple Recurrence Theorem for probability-preserving systems in order to give an ergodic-theoretic proof of Szemeredi's Theorem in additive combinatorics. In the thirty years since that work, the study of the multiple ergodic averages that underlie proofs of multiple recurrence has become a sophisticated theory in its own right. In addition to the positivity that implies multiple recurrence, their convergence and a description of their limits are of particular interest. I will discuss some recent work on the continuous-time multiple averages associated to a tuple of polynomial flows in an acting nilpotent Lie group. This work relies on the formulation of convergence and recurrence phenomena in terms of the equidistribution of certain self-joinings on a Cartesian power of the original system under an off-diagonal polynomial flow. I will emphasize two key ingredients in the proof of this equidistribution that seem to indicate a parallel with the study of unipotent flows on homogeneous spaces, although the technical details of the proofs remain quite different. The first ingredient is a kind of measure-classification that constrains the possible structure of any subsequential limit joinings, and builds on several older works studying characteristic factors'. The second ingredient is an auxiliary result promising that given a family of such off-diagonal flows parametrized by polynomials in some extra real parameters, a generic' flow in this family (suitably interpreted) gives equidistribution to a limit joining that is independent of the parameter and is invariant under the group of off-diagonal transformations generated by all the flows in the whole family: this amounts to a kind of Pugh-Shub phenomenon among joinings.