A Family of Markov Shifts (almost) Classified by Periodic Points

Let G be a finite group, and let XG = {x = (x(s,t)) ∈ GZ 2 | x(s,t) = x(s,t−1) · x(s+1,t−1) for all (s, t) ∈ Z}. The compact zero–dimensional set XG carries a natural shift Z2–action σG and the pair ΣG = ( XG, σ G ) is a two–dimensional topological Markov shift. Using recent work by Crandall, Dilcher and Pomerance on the Fermat quotient, we show the following: 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× 1012, and H is any abelian group for which ΣG has the same periodic point data as ΣH , then G is isomorphic to H. This result may be viewed as an example of the “rigidity” properties of higher– dimensional Markov shifts with zero entropy.