Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space
Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space
复制标题
射影空间的主导理性自映射的动态度、算术熵和规范高度
DOI：
10.1017/etds.2012.144
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发表时间：
2011
影响因子：
0.9
通讯作者：
J. Silverman
中科院分区：
文献类型：
--
作者：
J. Silverman
设φ:ℙN⤏ℙN是占优有理映射。φ的动态度是量δφ=Lim(degφn)1/n.当φ定义在${\bar{{\mathbb{q}$上时，对于适当选择的αφ，我们将点$P在{\mathbb{P}}^N({\bar{{\mathbb{q})$的算术次数定义为φ(P)=Lim sup h(varphi n(P))1/n，并将P的典范高度定义为$Hat{h}_varphi(P)=\limsup\Delta_\varphi^{-n}n^{-\ell_\varphi}h(\varphi^n(P))$。我们首先证明了一些初等关系，并对δφ，αφ(P)，φ(P)和P的轨道的Zariski密度作了一些深层次的猜想，然后证明了我们关于单项映射的猜想。
Abstract Let φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a point $P\in {\mathbb {P}}^N({\bar {{\mathbb {Q}}}})$ to be αφ(P)=lim sup h(φn(P))1/n and the canonical height of P to be $\hat {h}_\varphi (P)=\limsup \delta _\varphi ^{-n}n^{-\ell _\varphi }h(\varphi ^n(P))$ for an appropriately chosen ℓφ. We begin by proving some elementary relations and making some deep conjectures relating δφ, αφ(P) , ${\hat h}_\varphi (P)$, and the Zariski density of the orbit ?φ(P) of P. We then prove our conjectures for monomial maps.