Ward, Thomas B. Periodic points for expansive actions of \(\mathbb Z^ d\) on compact abelian groups. (English) Zbl 0725.22003 Bull. Lond. Math. Soc. 24, No. 4, 317-324 (1992). We show that the periodic points of an expansive \({\mathbb{Z}}^ d\) action on a compact abelian group are uniformly distributed with respect to Haar measure if the action has completely positive entropy. In the general expansive case, we show that any measure obtained as the distribution of periodic points along some sequence of periods necessarily has maximal entropy but need not be Haar measure. Reviewer: T.Ward (Columbus) Cited in 5 Documents MSC: 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 22D40 Ergodic theory on groups 37A15 General groups of measure-preserving transformations and dynamical systems 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 43A05 Measures on groups and semigroups, etc. 22C05 Compact groups 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions Keywords:periodic points; compact abelian group; uniformly distributed; Haar measure; completely positive entropy PDFBibTeX XMLCite \textit{T. B. Ward}, Bull. Lond. Math. Soc. 24, No. 4, 317--324 (1992; Zbl 0725.22003) Full Text: DOI