  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.
