Periods of rational maps modulo primes

Let K be a number field, let φ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace α by a hypersurface, such as the ramification locus of a morphism φ : Pn → Pn . R. L. Benedetto Department of Mathematics, Amherst College, Amherst, MA 01002, USA e-mail: [email protected] D. Ghioca Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada e-mail: [email protected] B. Hutz Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, NY 10016-4309, USA e-mail: [email protected] P. Kurlberg Department of Mathematics, KTH, 100 44 Stockholm, Sweden e-mail: [email protected] T. Scanlon Mathematics Department, University of California Berkeley, Evans Hall, Berkeley, CA 94720-3840, USA e-mail: [email protected] T. J. Tucker (B) Department of Mathematics, University of Rochester, Rochester, NY 14627, USA e-mail: [email protected]