Arithmetic degrees of dynamical systems over fields of characteristic zero

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.