Arithmetic degrees for dynamical systems over function fields of characteristic zero

TL;DR

This paper investigates the relationship between arithmetic and dynamical degrees of rational self-maps over function fields of characteristic zero, providing new proofs, conditions for equality, and detailed analysis on projective spaces.

Contribution

It offers a new proof of the inequality between arithmetic and dynamical degrees, establishes conditions for their equality, and analyzes these concepts specifically for maps on projective spaces.

Findings

Arithmetic degree is always less than or equal to the dynamical degree.

Conditions are identified under which the arithmetic degree equals the dynamical degree.

Many points have arithmetic degrees equal to the dynamical degree.

Abstract

We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are interpreted into geometric ones. We give another proof of the theorem that the arithmetic degree at any point is smaller than or equal to the dynamical degree. We give a sufficient condition for an arithmetic degree to coincide with the dynamical degree, and prove that any self-map has so many points whose arithmetic degrees are equal to the dynamical degree. We study dominant rational self-maps on projective spaces in detail.