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

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.