For more information about this meeting, contact Anatole Katok, Dmitri Burago.

Title: | Equidistribution of joinings under off-diagonal polynomial flows |

Seminar: | Center for Dynamics and Geometry Seminars |

Speaker: | Tim Austin, Brown University |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 02 / 29 / 2012 |

Time: | 03:35pm - 05:30pm |