Polynomial iteration in characteristic p

Let f (x) = d s=0 asxs ∈ Z[x] be a polynomial with ad = 0 mod p. Take z ∈ Fp and let Oz = {fi (z)} i∈Z+ ⊂ Fp be the orbit of z under f, where fi (z) = f (fi−1(z)) and f0(z) = z. For M <|Oz|, we study the diameter of the partial orbit O z,M = {z, f (z), f2(z), . . . , fM−1(z)} and prove that diamO z,M min Mc log logM,Mpc,M 12 p 12 , where ‘diameter’ is naturally defined in Fp and c depends only on d. For a complete orbit C, we prove that diam C min pc, eT/4 , where T is the period of the orbit. © 2012 Elsevier Inc. All rights reserved