Polynomial iteration in characteristic p ∗ †

Let f(x) = ∑d s=0 asx s ∈ Z[x] be a polynomial with ad 6≡ 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 Oz,M = {z, f(z), f2(z), . . . , fM−1(z)} and prove that diam Oz,M & min { M c log logM , Mp , M 1 2p 1 2 } , where ‘diameter’ is naturally defined in Fp and c depends only on d. For a complete orbit C, we prove that diam C & min{p , e T/4 }, where T is the period of the orbit. ∗2000 Mathematics Subject Classification.Primary 11B50, 37A45, 11B75; Secondary 11T23, 37F10 11G99. †