A bound for the sum of heights on iterates in terms of a dynamical degree

TL;DR

This paper establishes an upper bound for the sum of heights along iterates of a rational dynamical system on a projective variety, linking it to the system's dynamical degree and initial height.

Contribution

It provides a new bound relating the sum of heights of iterates to the dynamical degree and initial height, extending previous work on height growth in dynamical systems.

Findings

Bound holds for all points with well-defined orbits

The sum of heights grows at most exponentially with rate given by the dynamical degree

The result applies to any Weil height with respect to an ample divisor

Abstract

We give a proof for a fact that for any Weil height $h_X$ with respect to an ample divisor on a projective variety $X$, any dynamical system $\mathcal{F}$ of rational self-maps on $X$, and any $\epsilon>0$, there is a positive constant $C=C(X, h_X, f, \epsilon)$ such that $\sum_{f \in \mathcal{F}_n} h^+_X(f(P)) \leq C. k^n.(\delta_{\mathcal{F}} + \epsilon)^n . h^+_X(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined, with $\delta_{\mathcal{F}}$ being a dynamical degree associated with a system of several maps, defined by the author in the previous paper mentioned above.