Let K be a number field, let ϕ(x) ∈ K(x) be a rational function of degree d > 1, and let α ∈ K be a wandering point such that ϕ n (α) = 0 for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that v p (ϕ n (α)) > 0 and vp(ϕ m (α)) 0 for all positive integers m < n. Under appropriate ramification hypotheses, we can replace the condition v p (ϕ n (α)) > 0 with the stronger condition vp(ϕ n (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when ϕ is not isotrivial.

Abstract. We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let φ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of (P 1 ) g has only finite intersection with any curve contained in (P 1 ) g . Our proof uses results from p-adic dynamics together with an integrality argument.

Under suitable hypotheses, we prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism Φ : X → X.We also prove a version of the Mordell-Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.

We provide a family of counterexamples to a first formulation of the dynamical ManinMumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of Abelian varieties under the action of group endomorphisms and for lines under the action of diagonal endomorphisms of P 1 × P 1 .