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Periodic point data detects subdynamics in entropy rank one. (English) Zbl 1142.37012

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.

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B99 Topological dynamics
54H20 Topological dynamics (MSC2010)
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