2024/01/22 by Wenbin Luo, Luo, Wenbin, Jiarui Song +1 · 1 citation
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.11982
openalex publication_date 2024/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we generalize the arithmetic degree and its related theory to dynamical systems defined over an arbitrary field k of characteristic 0. We first consider a dynamical system (X,f) over a finitely generated field K over ℚ, we introduce the arithmetic degrees α(f,⋅) for 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 α(f,x) is less than or equal to the first dynamical degree λ1(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 k=ℂ, from which we deduce that α(f,x)=λ1(f) for very general x∈ X(ℂ) when f is an endomorphism.