American Journal of Mathematics

Abstract:

Let $X$ be a smooth projective variety defined over $\overline{\Bbb{Q}}$, and $f\colon X\dashrightarrow X$ be a dominant rational map. Let $\delta_f$ be the first dynamical degree of $f$ and $h_X\colon X(\overline{\Bbb{Q}})\rightarrow [1,\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X(f^n(P)))_{n\geq0}$ where $P$ is a point of $X(\overline{\Bbb{Q}})$ whose forward orbit by $f$ is well defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $\overline{\alpha}_f(P)\leq\delta_f$. Furthermore, we prove the canonical height functions of rational self-maps exist under certain conditions. For example, when the Picard number of $X$ is one, $f$ is algebraically stable (in the sense of Fornaess-Sibony) and $\delta_f>1$, the limit defining canonical height $\lim_{n\to\infty}h_X(f^n(P))\big/\delta_f^n$ converges.