EVENT DETAILS AND ABSTRACT

Title: Joint ergodicity for polynomial ergodic averages with commuting transformations
Speaker: Nikos Frantzikinakis
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Abstract:

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 worksby several mathematicians, their norm convergence was established in full generality by Walsh. But little has been known about their limit, even in seemingly simple cases of two commuting weakly mixing transformations with independent polynomial iterates. I will discuss recent joint work with B. Kuca where we manage to somewhat rectify the situation and answer several natural open problems. Our proof is based on two new techniques. The first is a ``degree lowering'' technique that leads to joint ergodicity criteria for ergodic averages with general iterates. The second is ``seminorm smoothing" technique that allows in several cases of interest to boost estimates involving box seminorms to estimates involving the Gowers-Host-Kra seminorms, which is a necessary requirement for the application of the degree lowering technique.