The Dynamical And Arithmetical Degrees For Eigensystems of Rational Self-maps

TL;DR

This paper introduces and studies arithmetical and dynamical degrees for systems of rational maps on projective varieties, establishing their properties, relations, and the existence of a canonical height function in characteristic zero.

Contribution

It defines new degrees for rational self-maps, proves their properties, and constructs a canonical height function linked to divisorial relations in the Néron-Severi group.

Findings

Existence of a canonical height function for morphisms over global fields.

Bound on the growth of Weil heights under iterated rational maps.

Relation between dynamical degrees and height growth rates.

Abstract

We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial relations in the N\'{e}ron-Severi Group over Global fields of characteristic zero, when the rational maps are morphisms. For such, we show that for any Weil height $h_X$ with respect to an ample divisor on a projective variety $X$, any dynamical system $\mathcal{F}$ of rational self-maps on $X$, and any $\epsilon>0$, there is a positive constant $C=C(X, h_X, f, \epsilon)$ such that $\sum_{f \in \mathcal{F}_n} h^+_X(f(P)) \leq C. k^n.(\delta_{\mathcal{F}} + \epsilon)^n . h^+_X(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined.