## 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)

### Keywords:

expansive action; expansive subdynamics; entropy
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