## 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.

### MSC:

 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

### Keywords:

periodic points; topological Markov shift; Wieferich prime

Zbl 0854.11002
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### References:

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