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

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 $ϵ>0$, there is a positive constant $C=C(X,h_X,f,ϵ)$ such that $\sum_{f \in $\mathcal{F}$_n} h^+X(f(P)) \leq C. k^n.(\delta{$\mathcal{F}$} + \epsilon)^n . h^+_X(P)$forallpoints$P$whose$$\mathcal{F}$$-orbit is well defined.