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arXiv · 1707.03943

The Dynamical And Arithmetical Degrees For Eigensystems of Rational Self-maps

Jorge Mello
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Abstract

We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial relations in the N\'{e}ron-Severi Group over Global fields of characteristic zero, when the rational maps are morphisms. For such, we show that for any Weil height $h_X$ with respect to an ample divisor on a projective variety $X$, any dynamical system $\mathcal{F}$ of rational self-maps on $X$, and any $\epsilon>0$, there is a positive constant $C=C(X, h_X, f, \epsilon)$ such that $\sum_{f \in \mathcal{F}_n} h^+_X(f(P)) \leq C. k^n.(\delta_{\mathcal{F}} + \epsilon)^n . h^+_X(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined.

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Jorge Mello. 2017-07-13. The Dynamical And Arithmetical Degrees For Eigensystems of Rational Self-maps. https://arxiv.org/abs/1707.03943

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