×

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

Citations:

Zbl 0854.11002
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Crandall, R.; Dilcher, K.; Pomerance, C., A search for wieferich and Wilson primes, Math. comp., 66, 433-449, (1997) · Zbl 0854.11002
[2] Ledrappier, F., Un champ markovien peut être d’entropie nulle et melangeant, Comptes rendus acad. sci. Paris, ser. A., 287, 561-562, (1978) · Zbl 0387.60084
[3] Pinch, R.G.E., Recurrent sequences modulo prime powers, Cryptography and coding III, (1993), Oxford Univ. Press Oxford · Zbl 0822.11012
[4] Ribenboim, P., “1093”, Math. intelligencer, 5, 28-33, (1983) · Zbl 0516.10001
[5] Ribenboim, P., The book of prime number records, (1988), Springer-Verlag New York · Zbl 0642.10001
[6] Schmidt, K., Algebraic ideas in ergodic theory, C.B.M.S. reg. conf. ser. in math., 76, (1990)
[7] Schmidt, K., Dynamical systems of algebraic origin, (1995), Birkhaüser Basel · Zbl 0833.28001
[8] Shereshevsky, M.A., On the classification of some two-dimensional Markov shifts with group structure, Ergod. th. and dyn. sys., 12, 823-833, (1992) · Zbl 0781.58017
[9] Ward, T.B., Almost block independence for the three dotZ2, Israel J. of math., 76, 237-256, (1991) · Zbl 0790.28013
[10] Ward, T.B., Periodic points for expansive actions ofZ^{d}on compact abelian groups, Bull. London math. soc., 24, 317-324, (1992) · Zbl 0725.22003
[11] Ward, T.B., An algebraic obstruction to isomorphism of Markov shifts with group alphabets, Bull. London math. soc., 25, 240-246, (1993) · Zbl 0792.22004
[12] Wieferich, A., Zum letzten fermatschen theorem, J. für die reine und angewandte math., 136, 293-302, (1909) · JFM 40.0256.03
[13] Zassenhaus, H.J., The theory of groups, (1949), Chelsea New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.