Arithmetic degrees of dynamical systems over fields of characteristic zero

Abstract

In this article, we generalize the arithmetic degree and its related theory to dynamical systems defined over an arbitrary field $\mathbf{k}$ of characteristic $0$. We first consider a dynamical system $(X,f)$ over a finitely generated field $K$ over $\mathbb{Q}$, we introduce the arithmetic degrees $\alpha(f,\cdot)$ for $\overline{K}$-points by using Moriwaki heights. We study the arithmetic dynamical degree of $(X,f)$ and establish the relative degree formula. The relative degree formula gives a proof of the fundamental inequality, that is, the upper arithmetic degree $\overline{\alpha}(f,x)$ is less than or equal to the first dynamical degree $\lambda_1(f)$ in this setting. By taking spread-outs, we extend the definition of arithmetic degrees to dynamical systems over the field $\mathbf k$. We demonstrate that our definition is independent of the choice of the spread-out. Moreover, in this setting, we prove certain special cases of the Kawaguchi-Silverman conjecture. A main novelty of this paper is that, we give a characterization of arithmetic degrees of "transcendental points" in the case $\mathbf{k}=\mathbb{C}$, from which we deduce that $\alpha(f,x)=\lambda_1(f)$ for very general $x\in X(\mathbb{C})$ when $f$ is an endomorphism.