Multiple recurrence and convergence without commutativity

Nikos Frantzikinakis, Bernard Host

Published 2021-11-02, updated 2023-01-11Version 3

We establish multiple recurrence and convergence results for pairs of zero entropy measure preserving transformations that do not satisfy any commutativity assumptions. Our results cover the case where the iterates of the two transformations are $n$ and $n^k$ respectively, where $k\geq 2$, and the case $k=1$ remains an open problem. Our starting point is based on the observation that Furstenberg systems of sequences of the form $(f(T^{n^k}x))$ have very special structural properties when $k\geq 2$. We use these properties and some disjointness arguments in order to get characteristic factors with nilpotent structure for the corresponding ergodic averages, and then finish the proof using some equidistribution results on nilmanifolds.