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

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 Néron-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 ${\sum }_{f\in {\mathcal{F}}_n}{h}_X^{+}(f(P))\le C.k^n.({\delta }_{\mathcal{F}}+\epsilon {)}^n.h_X^{+}(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined.