  For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ where
we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we
study the pointwise convergence of multiple ergodic averages with iterates of
different growth coming from a large class of sublinear functions. This class
properly contains important subclasses of Hardy field functions of order $0$
and of Fej\'er functions, i.e., tempered functions of order $0.$ We show that
the convergence of the single average, via an invariant property, implies the
convergence of the multiple one. We also provide examples of sublinear
functions which are in general bad for convergence on arbitrary systems, but
they are good for uniquely ergodic systems. The case where the fastest function
is linear is addressed as well, and we provide, in all the cases, an explicit
formula of the limit function.
