Periods of rational maps modulo primes

Abstract

Let K be a number field, let $${\varphi \in K(t)}$$ be a rational map of degree at least 2, and let $${\alpha, \beta \in K}$$ . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes $${\mathfrak{p}}$$ of K such that $${\alpha {\rm mod} \mathfrak{p}}$$ is not in the forward orbit of $${\beta {\rm mod} \mathfrak{p}}$$ . 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 $${\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}}$$ .