Arithmetic and Dynamical Degrees on Abelian Varieties

Let phi : X -> X be a dominant rational map of a smooth variety and let x is an element of X, all defined over (Q) over bar. The dynamical degree delta(phi) measures the geometric complexity of the iterates of 0, and the arithmetic degree alpha(phi, x) measures the arithmetic complexity of the forward phi-orbit of x. It is known that alpha(phi, x) <= delta(phi), and it is conjectured that if the phi-orbit of x is Zariski dense in X, then alpha(phi, x) = delta(phi), i.e. arithmetic complexity equals geometric complexity. In this note we prove this conjecture in the case that X is an abelian variety, extending earlier work in which the conjecture was proven for isogenies.