The Dynamical and Arithmetical Degrees for Eigensystems of Rational Self-maps

  • Jorge MelloEmail author


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.


Canonical heights Dynamical degree Arithmetic degree Néron-Severi group Preperiodic rational points 



The author was supported by CAPES, ARC Discovery Grant DP180100201 and UNSW in this research.


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© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics UNSWUniversity of New South WalesSydneyAustralia

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