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 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 $h_X^+ = \max \{1, h_X\}$ with respect to an ample divisor on a smooth projective variety X, any dynamical system ${\mathcal {F}}$ of rational self-maps on X with dynamical degree $\delta _{{\mathcal {F}}}$, ${\mathcal {F}}_n$ its set of $n-$iterates, and any $\epsilon >0$, there is a positive constant $C=C(X, h_X, {\mathcal {F}}, \epsilon )$ such that

$$\begin{aligned} \mathop \sum \limits _{f \in {\mathcal {F}}_n} h^+_X(f(P)) \le C. k^n.(\delta _{{\mathcal {F}}} + \epsilon )^n . h^+_X(P) \end{aligned}$$

for all points P whose ${\mathcal {F}}$-orbit is well defined.