Miles, Richard; Ward, Thomas B. Periodic point data detects subdynamics in entropy rank one. (English) Zbl 1142.37012 Ergodic Theory Dyn. Syst. 26, No. 6, 1913-1930 (2006). Let \(\beta\) be an action of \(\mathbb{Z}^d\) by homeomorphisms of a compact metric space \((X,\rho)\), therefore for any \(n\in\mathbb{Z}^d\) there exists an homeomorphism \(\beta^n\) and it holds \(\beta^n\circ\beta^m=\beta^{n+m}\) for all \(n,m\in\mathbb{Z}^d.\) This action is called expansive if there exists some \(\delta>0\) such that if \(x,y\) are distinct points in \(X\), then there is some \(n\) for which \(\rho(\beta^nx,\beta^ny)>\delta.\) Let \(F_n(\beta)=\{x\in X: \beta^nx=x\}\) and \(C:\mathbb{Z}^d\to \mathbb{N}\cup\{\infty\}\) the map such that \(C(n)\) is the cardinal of \(F_n(\beta)\). The authors prove that the combinatorial data contained in the map \(C\) determine the expansive subdynamics of the expansive algebraic system of entropy rank one. In particular, for these systems the set \(F_n(\beta)\) is finite for \(n\not=0\) except in degenerate situations. Reviewer: Gabriel Soler López (Cartagena) Cited in 4 Documents MSC: 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37B99 Topological dynamics 54H20 Topological dynamics (MSC2010) Keywords:expansive action; expansive subdynamics; entropy PDFBibTeX XMLCite \textit{R. Miles} and \textit{T. B. Ward}, Ergodic Theory Dyn. Syst. 26, No. 6, 1913--1930 (2006; Zbl 1142.37012) Full Text: DOI arXiv