  We study measure preserving systems, called Furstenberg systems, that model
the statistical behavior of sequences defined by smooth functions with at most
polynomial growth. Typical examples are the sequences $(n^\frac{3}{2})$,
$(n\log{n})$, and $([n^\frac{3}{2}]\alpha)$, $\alpha\in
\mathbb{R}\setminus\mathbb{Q}$, where the entries are taken $\mod{1}$. We show
that their Furstenberg systems arise from unipotent transformations on finite
dimensional tori with some invariant measure that is absolutely continuous with
respect to the Haar measure and deduce that they are disjoint from every
ergodic system. We also study similar problems for sequences of the form
$(g(S^{[n^{\frac{3}{2}}]} y))$, where $S$ is a measure preserving
transformation on the probability space $(Y,\nu)$, $g\in L^\infty(\nu)$, and
$y$ is a typical point in $Y$. We prove that the corresponding Furstenberg
systems are strongly stationary and deduce from this a multiple ergodic theorem
and a multiple recurrence result for measure preserving transformations of zero
entropy that do not satisfy any commutativity conditions.
