Let ϕ:V→ V be a self-morphism of a quasiprojective variety defined over a number field K and let P∈ V(K) be a point with infinite orbit under iteration of ϕ. For each prime p of good reduction, let m_p(ϕ,P) be the size of the ϕ-orbit of the reduction of P modulo p. Fix any ε>0. We show that for almost all primes p in the sense of analytic density, the orbit size m_p(ϕ,P) is larger than (log N_K/Qp)^1-ε.

The author's research supported by NSF grant DMS-0650017