A family of Markov shifts (almost) classified by periodic points. (English) Zbl 0935.28008

Let \(G\) be a finite group and consider the set \[ X_G= \{{\mathbf x}= (x_{(s,t)})\in G^{\mathbb Z^2}: x_{(s,t)}= x_{(s,t-1)}\cdot x_{(s+ 1,t-1)},(s, t)\in\mathbb Z^2\}. \] The set \(X_G\) inherits a natural shift \(\mathbb Z^2\)-action \(\sigma= \sigma^G\), so that the pair \(\Sigma_G= (X_G,\sigma^G)\) is a two-dimensional topological Markov shift. Let \(\Gamma\) be a finite subgroup of \(\mathbb Z^2\) with finite index, the set of \(\Gamma\)-periodic points is defined as follows \[ \text{Fix}_\Gamma(\sigma)= \{{\mathbf x}\in X_G: \sigma_{{\mathbf n}}(x)= x,\text{ for all }{\mathbf n}\in\Gamma\}. \] Denote \(|\text{Fix}_\Gamma(\sigma)|\) by \(F_\Gamma(\sigma)\), which is an invariant of topological conjugacy. Using recent work by R. Crandall, K. Dilcher and C. Pomerance on the Fermat quotient [Math. Comput. 66, No. 217, 433–449 (1997; Zbl 0854.11002)], the author proves that if \(G\) is abelian, and the order of \(G\) is not divisible by 1024, nor by the square of any Wieferich prime larger than \(4\times 10^{12}\), and \(H\) is any abelian group for which \(\Sigma_G\) has the same periodic point data as \(\Sigma_G\), then \(G\) is isomorphic to \(H\). The following conjecture remains open: if \(G\) and \(H\) are finite abelian groups, and \(F_\Gamma(\sigma^G)= F_\Gamma(\sigma^H)\) for all periods \(\Gamma\), then \(G\) and \(H\) are isomorphic. The author claims that a proof of this conjecture requires a different approach than the one used in this paper.


37B99 Topological dynamics
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
28D20 Entropy and other invariants
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
54H20 Topological dynamics (MSC2010)
37E05 Dynamical systems involving maps of the interval


Zbl 0854.11002
Full Text: DOI Link


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