  In this paper we study a class of dynamical systems generated by iterations
of multivariate permutation polynomial systems which lead to polynomial growth
of the degrees of these iterations. Using these estimates and the same
techniques studied previously for inversive generators, we bound exponential
sums along the orbits of these dynamical systems and show that they admit much
stronger estimates on average over all initial values than in the general case
and thus can be of use for pseudorandom number generation.
