Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps

TL;DR

This paper investigates the behavior of arithmetic degrees and Zariski dense orbits for cohomologically hyperbolic maps on projective varieties, establishing key results about the equality of arithmetic and dynamical degrees and the existence of points with dense orbits.

Contribution

It proves that for 1-cohomologically hyperbolic maps, the arithmetic degree of generic points equals the first dynamical degree and confirms the existence of points with dense orbits, including transcendental cases.

Findings

Arithmetic degree equals the first dynamical degree for generic points.

Existence of points with Zariski dense orbits.

Transcendental dynamical degrees can occur in arithmetic degrees.

Abstract

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