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

In this talk, for an ergodic probability preserving system (X,B,m,T), we will discuss the existence of a Z^d valued function , 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. We also provide first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff.