The dynamics of linearized polynomials

Let F = GF(q). To any polynomial G is an element of F[x] there is associate d a mapping (G) over cap on the set I-F of monic irreducible polynomials ov er F. We present a natural and effective theory of the dynamics of (G) over cap for the case in which G is a monic q-linearized polynomial. The main o utcome is the following theorem. Assume that G is not of the form x(ql), where l greater than or equal to 0 (in which event the dynamics is trivial). Then, for every integer n greater than or equal to 1 and for every integer k greater than or equal to 0, the re exist infinitely many mu is an element of I-F having preperiod k and pri mitive period n with respect to (G) over cap. Previously, Morton, by somewhat different means, had studied the primitive periods of (G) over cap when G = x(q) - ax, a a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.