Limit Formulas for Polynomial Ergodic Averages with Commuting Transformations

The works of Furstenberg and Bergelson-Leibman on the Szemeredi theorem and its polynomial extension motivated the study of the limiting behavior of multiple ergodic averages of commuting transformations with polynomial iterates. Following important works of Conze-Lesigne, Host-Kra, Ziegler, Tao, and others, their norm convergence was established in full generality by Walsh. But little has been known about their limit, even in the seemingly simple case of two commuting weakly mixing transformations with iterates given by independent polynomials.

I will discuss recent joint work with B. Kuca, where we use a new approach to somewhat rectify the situation and answer several natural open problems.