Topological Semi-Conjugacies Between Adding Machines

  • Jie-Hua Mai
  • Cui-Jun Liu
  • Xin-He Liu


Let K = (k 1, k 2, . . .) be a sequence of positive integers, and \({\Sigma_{K} = \prod_{n=1}^\infty {\bf Z}_{k_n}}\) be a topological group with a metric and an additive operation given. The adding machine f K is the addition-by-one map on Σ K . Buescu and Stewart (Ergodic Theory Dynam Syst 15:271–290, 1995), Block and Keesling (Topology Appl 140:151–161, 2004) and Banks (Ergodic Theory Dynam Syst 17:505–529, 1997) obtained several equivalent conditions for which two adding machines are topologically conjugate, respectively. In this paper, we have a further discussion about semi-conjugate relationship between adding machines, and give some necessary and sufficient conditions for an adding machine to be semi-conjugate to another one. Moveover, we also prove that a transitive translation on a compact subgroup of Σ K is isometrically conjugate to an adding machine.


Adding machine Topological group Homomorphism Topological conjugacy Topological semi-conjugacy 

Mathematics Subject Classification (2000)

Primary 37B10 37C15 Secondary 54E40 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of MathematicsShantou UniversityShantou, GuangdongPeople’s Republic of China
  2. 2.School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China
  3. 3.Institute of MathematicsGuangxi UniversityNanning, GuangxiPeople’s Republic of China

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