Let g and h be monic polynomials in F[x], where F is the finite field of order q. We define a dynamical system by letting the q-linearized polynomial associated with g act on equivalence classes of a certain F-subspace of the algebraic closure of F in which related elements of the closure lie in the same orbit under the action of the q-linearized polynomial associated with h. When h = x, this is equivalent to the system in which the dynamic polynomial g acts on irreducible polynomials over F as discussed in [CH], where a conjecture of Morton [M] was proved as regards linearized polynomials. A generalization of that result is proved here. This states that when g and h are non-constant relatively prime polynomials, then there are infinitely many classes with prescribed preperiod and primitive period in the (g,h)-dynamical system.