Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

TL;DR

This paper proves a multidimensional local limit theorem for ergodic systems and applies it to demonstrate non-convergence of polynomial multiple averages and failure of multiple recurrence in certain zero entropy systems.

Contribution

It establishes a multidimensional local limit theorem in deterministic systems and uses it to show non-convergence of polynomial multiple averages and recurrence failure.

Findings

Existence of cocycles satisfying the d-dimensional local CLT in ergodic systems

Counterexamples to convergence of polynomial multiple averages in zero entropy systems

First examples of failure of multiple recurrence along polynomial iterates in such systems

Abstract

We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the $2$-dimensional result to 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. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.