论文标题
算术学学位和Zariski密集的杂质图
Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps
论文作者
论文摘要
如果$ p $ th的动力学学位严格比其他动力学学位大，则在投影品种上的主要理性自我图称为$ P $ - 生物学上的双曲线。对于在$ \ overline {\ mathbb {q}} $上定义的地图，我们研究了算术学度的下限，存在Zariski密集轨道的点以及前观点的有限度。特别是，我们证明，如果$ f $是光滑的投影品种上的$ 1 $ - 生物学上的双曲线图，则（1）$ \ OVERLINE {\ MATHBB {Q}} $的算术度与通用$ f $ -orbit的指数相当于第一个动态度$ f $ $ f $; （2）有$ \ overline {\ mathbb {q}} $ - 带有通用$ f $ -orbit的点。将我们的定理应用于具有先验动力学程度的最近构建的有理图，我们确认算术程度可以先验。
A dominant rational self-map on a projective variety is called $p$-cohomologically hyperbolic if the $p$-th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over $\overline{\mathbb{Q}}$, we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if $f$ is an $1$-cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a $\overline{\mathbb{Q}}$-point with generic $f$-orbit is equal to the first dynamical degree of $f$; and (2) there are $\overline{\mathbb{Q}}$-points with generic $f$-orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental.
