Growth rate of ample heights and the dynamical Mordell-Lang type conjecture

TL;DR

This paper derives an explicit formula for the growth rate of ample heights under endomorphisms on smooth projective varieties and applies it to a dynamical Mordell-Lang type problem involving étale endomorphisms.

Contribution

It provides a new explicit formula for height growth rates and solves a variant of the dynamical Mordell-Lang conjecture for étale endomorphisms.

Findings

Explicit formula for height growth rate under endomorphisms

Positive solution to a dynamical Mordell-Lang variant

Application to pairs of étale endomorphisms

Abstract

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms on smooth projective varieties over number fields. As an application, we give a positive answer to a problem of Dynamical Mordell-Lang type for pairs of \'etale endomorphisms, which is a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.