The Collatz problem in F_p[x] and F_p[[x]]

We study the analogue of the Collatz map in the polynomial ring F_p[x], for any prime number p, and the corresponding dynamical system. We show that every f∈F_p[x] is eventually periodic in this system, in a quadratic number of iterations in deg(f), and describe explicitly all corresponding cycles. This extends a result of Hicks, Mullen, Yucas and Zavislak, who studied the case p=2. We also study the Collatz map in the formal power series ring F_p[[x]], observe that in F_p[[x]] all but countably many power series generate divergent trajectories via iterations of this map, and characterize those power series that are eventually periodic.