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… (More)
Let ϕ : X → X be a morphism of a variety over a number field K. We consider local conditions and a " Brauer-Manin " condition, defined by Hsia and Silverman, for the orbit of a point P ∈ X(K) to be disjoint from a subvariety V ⊆ X, i.e., for O ϕ (P) ∩ V = ∅. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of… (More)
We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y +1/y) = n as n → ∞. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.