Zariski Density of Points with Maximal Arithmetic Degree

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 paper, we show that there are densely many $\bar{\mathbb{Q}}$-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.