On upper bounds of arithmetic degrees

Abstract

Let $X$ be a smooth projective variety over $\overline{\mathbb{Q}}$, and $f:X-rightarrowX$ be a dominant rational map. Let ${\delta }_f$ be the first dynamical degree of $f$ and $h_X:X(\overline{\mathbb{Q}})\to [1,\infty )$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X(f^n(P)){)}_{n\ge 0}$ where $P$ is a point of $X(\overline{\mathbb{Q}})$ whose forward orbit by $f$ is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; ${\overline{\alpha }}_f(P)\le {\delta }_f$. Furthermore, if the Picard number of $X$ is one, $f$ is algebraically stable and ${\delta }_f>1$, we prove that the limit defining canonical height ${\lim }_{n\to \infty }{h}_X(f^n(P))/{\delta }_f^n$ converges.

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• Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties
• Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties
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• Zariski density of points with maximal arithmetic degree for surfaces