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

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.