Polynomial iteration in characteristic p

Let f(x)=∑s=0dasxs∈Z[x] be a polynomial with ad≠0modp. Take z∈Fpz∈Fp and let Oz={fi(z)}i∈Z+⊂FpOz={fi(z)}i∈Z+⊂Fp be the orbit of z under f  , where fi(z)=f(fi−1(z))fi(z)=f(fi−1(z)) and f0(z)=zf0(z)=z. For M<|Oz|M<|Oz|, we study the diameter of the partial orbit Oz,M={z,f(z),f2(z),…,fM−1(z)}Oz,M={z,f(z),f2(z),…,fM−1(z)} and prove thatdiamOz,M≳min{McloglogM,Mpc,M12p12}, where ‘diameter’ is naturally defined in FpFp and c depends only on d  . For a complete orbit CC, we prove thatdiamC≳min{pc,eT/4},diamC≳min{pc,eT/4}, where T is the period of the orbit.