Multiple recurrence and convergence without commutativity

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