Georgia Institute of Technology College of Sciences

In this talk, for an ergodic probability preserving system $(X,\mathcal{B},m,T)$, we will discuss the existence of a function $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem.
As an application, we resolve a question of Huang, Shao and Ye, and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. If time allows, we will also discuss the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff (arXiv:2409.05087).