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 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.

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

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Correspondence to Jorge Mello.

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The author would like to say thanks to J. Silverman and S. Kawaguchi.

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Mello, J. The Dynamical and Arithmetical Degrees for Eigensystems of Rational Self-maps. Bull Braz Math Soc, New Series 51, 569–596 (2020).

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