Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps

Abstract

A dominant rational self-map on a projective variety is called p p -cohomologically hyperbolic if the p p -th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over Q ¯ \overline {\mathbb {Q}} , we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if f f is an 1 1 -cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a Q ¯ \overline {\mathbb {Q}} -point with generic f f -orbit is equal to the first dynamical degree of f f ; and (2) there are Q ¯ \overline {\mathbb {Q}} -points with generic f f -orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental.