On upper bounds of arithmetic degrees
Yohsuke Matsuzawa
Abstract
Let X be a smooth projective variety defined over Q, and f : X X be a dominant rational map. Let f be the first dynamical degree of f and h X : X(Q) - [1, ) 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 ))) n0 where P is a point of X(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; f (P ) 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 and f > 1, the limit defining canonical height limn h X (f n (P )) n f converges.