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 Neron-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.