We define arithmetical and dynamical degrees for dynamical systems with several rational maps on smooth 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 hX+=max{1,hX}\documentclass[12pt]{minimal}
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\begin{document}$$h_X^+ = \max \{1, h_X\}$$\end{document} with respect to an ample divisor on a smooth projective variety X, any dynamical system F\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}$$\end{document} of rational self-maps on X with dynamical degree δF\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{{\mathcal {F}}}$$\end{document}, Fn\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}_n$$\end{document} its set of n-\documentclass[12pt]{minimal}
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\begin{document}$$n-$$\end{document}iterates, and any ϵ>0\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon >0$$\end{document}, there is a positive constant C=C(X,hX,F,ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$C=C(X, h_X, {\mathcal {F}}, \epsilon )$$\end{document} such that ∑f∈FnhX+(f(P))≤C.kn.(δF+ϵ)n.hX+(P)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathop \sum \limits _{f \in {\mathcal {F}}_n} h^+_X(f(P)) \le C. k^n.(\delta _{{\mathcal {F}}} + \epsilon )^n . h^+_X(P) \end{aligned}$$\end{document}for all points P whose F\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}$$\end{document}-orbit is well defined.
