Dynamical Uniform Bounds for Fibers and a Gap Conjecture

Abstract

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our first result, we assume X is a quasi-projective variety defined over a field K of characteristic 0, endowed with the action of an étale endomorphism Φ, and f : X −→ Y is a morphism with Y a quasi-projective variety defined over K. Then for any x ∈ X(K), if for each y ∈ Y (K), the set Sy := {n ∈ N : f (Φ n (x)) = y} is finite, then there exists a positive integer N such that #Sy ≤ N for each y ∈ Y (K). For our second result, we let K be a number field, f : X P 1 is a rational map, and Φ is an arbitrary endomorphism of X. If OΦ(x) denotes the forward orbit of x under the action of Φ, then either f (OΦ(x)) is finite, or lim sup n→∞ h(f (Φ n (x)))/ log(n) > 0, where h(•) represents the usual logarithmic Weil height for algebraic points.