局部域上高维射影空间态射的动力性质

Dynamical Properties of Morphisms on Higher Dimensional Projective Space over Local Field

设Cv是关于非阿绝对值完备的代数闭域,φ:P^(N)→P^(N)为定义在Cv上次数大于1的态射,Φ是φ的提升,GΦ是Φ的Green函数,ρ是P^(N)(Cv)上的弦度量.本文首先通过弦度量研究了高维射影空间中点的约化及线性分式约化的性质.利用Green函数0Φ定义了态射提升的算术距离,研究了算术距离的性质,给出了Φ具有好的约化的充要条件.同时,利用Green函数明确刻画了Φ的Filled Julia集.

Let C_(v) be an algebraically closed non-archimedean field,complete with respect to a valuation v.Let φ:P^(N)→P^(N) be a morphism of degree greater than one defined over Cv,Φ a lift of φ.Let gΦ be the Green function of Φ and ρ the chordal metric on P^(N)(C_(v)).In this paper,we first study the properties of reduction of points in high dimensional projective space and reduction of automorphisms of P^(N) with degree one.With the help of Green function gΦ of Φ,we introduce the arithmetic distance of morphisms and investigate its property.The necessary and sufficient condition which Φ has good reduction is obtained in this paper.We also describe explicitly the Filled Julia set of Φ by its Green function.