Abstract
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.
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Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday
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Mai, JH., Liu, CJ. & Liu, XH. Topological Semi-Conjugacies Between Adding Machines. J Dyn Diff Equat 20, 883–892 (2008). https://doi.org/10.1007/s10884-008-9110-9
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DOI: https://doi.org/10.1007/s10884-008-9110-9