Zariski density of points with maximal arithmetic degree

TL;DR

This paper proves that for certain algebraic varieties with self-maps, points with maximal arithmetic degree are densely distributed, extending previous results and providing new insights into the distribution of such points over number fields.

Contribution

It establishes the density of points with maximal arithmetic degree for self-morphisms on projective, unirational, and abelian varieties, generalizing prior work and including a new appendix.

Findings

Densely many points with maximal arithmetic degree exist on projective varieties.

Such points are dense over large enough number fields for unirational and abelian varieties.

The paper generalizes a result of Kawaguchi and Silverman.

Abstract

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this article, we show that there are densely many $\overline{\mathbb Q}$-rational points with maximal arithmetic degree (i.e. whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the appendix.