Polynomial iteration in characteristic p

Let integral = Sigma(d)(s=0) a(s)x(s) is an element of Z[x] be a polynomial with a(d) not equal 0 mod p. Take z is an element of F-p and let O-z = {integral(i) (z)}(i is an element of z+) C F-p be the orbit of z under integral, where integral(i)(z) = integral(integral(i-1) (z)) and integral(0)(z) = z. For M < vertical bar O-z vertical bar, we study the diameter of the partial orbit O-z,(M) = {z, integral (z), integral(2)(z), . . . , integral (M-1)(z)} and prove that diam O-z, (M) greater than or similar to min {M-c log log M, Mp(c), M 1/2 p 1/2}, where 'diameter' is naturally defined in F-p and c depends only on d. For a complete orbit C, we prove that diam C greater than or similar to min {p(c), e(T/4)}, where T is the period of the orbit. (c) 2012 Elsevier Inc. All rights reserved.