Acta mathematica scientia,Series A ›› 2025, Vol. 45 ›› Issue (2): 305-320.

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Affine Semigroup Dynamical Systems on Zp

Xufei Lu1(),Changhua Jiao2(),Jinghua Yang1,*()   

  1. 1College of Science, Shanghai University, Shanghai 200444
    2Department of Mathematical Sciences, Tsinghua University, Beijing 100084
  • Received:2023-11-23 Revised:2024-10-15 Online:2025-04-26 Published:2025-04-09
  • Contact: Jinghua Yang E-mail:luxufei@shu.edu.cn;jch23@mails.tsinghua.edu.cn;jhyang@shu.edu.cn
  • Supported by:
    NSFC(12371073)

Abstract:

Let p2 be a prime and Zp be the ring of p-adic integers. For any α,β,zZp, define fα,β(z)=αz+β. The first part of this paper studies all minimal subsystems of semigroup dynamical systems (Zp,G) when fα1,β1 and fα2,β2 are commutative, where the semigroup G={fnα1,β1fmα2,β2:m,nN}. In particular, we find the semigroup dynamical system (Zp,G) (p3) is minimal if and only if (Zp,fα1,β1) or (Zp,fα2,β2) is minimal and we determine all the cases that (Z2,G) is minimal. In the second part, we study weakly essentially minimal affine semigroup dynamical systems on Zp, which is a kind of minimal semigroup systems without any minimal single action. It is shown that such semigroup is non-commutative when p3. Moreover, for a fixed prime p, we find the least number of generators of a weakly essentially minimal affine semigroup on Zp. 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.

Key words: minimal subsystem, p-adic dynamical system, affine semigroup

CLC Number: 

  • O19
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