Zp上仿射半群的动力系统  

Affine Semigroup Dynamical Systems on Z_(p)

作　　者：卢旭飞 焦昌华 杨静桦[1] Lu Xufei;Jiao Changhua;Yang Jinghua(College of Science,Shanghai University,Shanghai 200444;Department of Mathematical Sciences,Tsinghua University,Beijing 10084)

机构地区：[1]上海大学理学院,上海200444 [2]清华大学数学科学学院,北京100084

出　　处：《数学物理学报(A辑)》2025年第2期305-320,共16页Acta Mathematica Scientia

基　　金：国家自然科学基金(12371073)。

摘　　要：令p≥2为一素数,Zp为p-adic整数环.对任意的α,β,z∈Z_(p),定义f_(α,β)(z)=αz+β.该文第一部分研究了当f_(α1,β1)和f_(α2,β2)交换时的半群动力系统(Z_(p),G)的所有极小块,这里半群G={f_(α1,β1)^(n)○f_(α2,β2)^(m):m,n∈N}.特别地,我们找出了(Z_(p),G)(p≥3)是极小系统的充要条件是系统(Z_(p),f_(α1,β1))或(Z_(p),f_(α2,β2))极小并且找出了(Z_(2),G)是极小的所有情况.第二部分,考察了Z_(p)上的弱本质极小的仿射半群动力系统,这是一类半群中每个作用都不具有极小性但整体具有极小性的仿射系统.我们证明了:p≥3时这样的半群一定是非交换的.更进一步,给定素数p,我们想知道Z_(p)上弱本质极小仿射半群的生成元个数最少是多少.我们已经证明p=2和p=3时答案分别是2和3,对于一般的p,我们证明了这个数不超过p.Let p≥2 be a prime and Z_(p) be the ring of p-adic integers.For any α,β,z∈Z_(p),define f_(α,β)(z)=αz+β.The first part of this paper studies all minimal subsystems of semigroup dynamical systems(Z_(p),G)when f_(α_(1),β_(1))and f_(α_(2),β_(2))are commutative,where the semigroup G={f_(α1,β1)^(n)○f_(α2,β2)^(m):m,n∈N}.In particular,we find the semigroup dynamical system(Z_(p),G)(p≥3)is minimal if and only if(Z_(p),f_(α_(1),β_(1)))or(Z_(p),f_(α_(2),β_(2)))is minimal and we determine all the cases that(Z_(2),G)is minimal.In the second part,we study weakly essentially minimal affine semigroup dynamical systems on Z_(p),which is a kind of minimal semigroup systems without any minimal single action.It is shown that such semigroup is non-commutative when p≥3.Moreover,for a fixed prime p,we find the least number of generators of a weakly essentially minimal affine semigroup on Z_(p).We show that such number is 2 for p=2 and 3 for p=3.Also,we show that such number is not greater than p.

关 键 词：极小块 p-adic动力系统 仿射半群 

分 类 号：O19[理学—数学]