(Relative) dynamical degrees of rational maps over an algebraic closed field

TL;DR

This paper introduces the concept of dynamical degrees for rational maps over algebraically closed fields of characteristic zero, establishing their fundamental properties and applications, including a product formula for semi-conjugate maps.

Contribution

It defines relative dynamical degrees in the algebraic setting and proves a product formula, extending the theory to rational maps over algebraically closed fields.

Findings

Dynamical degrees satisfy log-concavity.

A product formula for semi-conjugate rational maps is established.

Results apply to surfaces and threefolds over fields of positive characteristic.

Abstract

The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define relative dynamical degrees and prove a "product formula" for dynamical degrees of semi-conjugate rational maps in the algebraic setting. The main tools are the Chow's moving lemma and a formula for the degree of the cone over a subvariety of $\mathbb{P}^N$. The proofs of these results are valid as long as resolution of singularities are available (or more generally if appropriate birational models of the maps under consideration are available). This observation is applied for the cases of surfaces and threefolds over a field of positive characteristic.