2019/06/20 by Bell, Jason, Ghioca, Dragos, Satriano, Matthew
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1906.08683
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\colon 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\colon f(Φn(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\dashrightarrow ℙ1 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 \limsupn→∞ h(f(Φn(x)))/log(n)>0, where h(⋅) represents the usual logarithmic Weil height for algebraic points.