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

Abstract

LetGbe a finite group, and letXG={x=(x(s, t))∈GZ2∣x(s, t)=x(s, t−1)·x(s+1, t−1)for all (s, t)∈Z2}.The compact zero-dimensional setXGcarries a natural shift Z2-actionσGand the pairσG=(XG, σG) is a two-dimensional topological Markov shift. Using recent work by Crandall, Dilcher and Pomerance on the Fermat quotient, we show the following: ifGis abelian, and the order ofGis not divisible by 1024, nor by the square of any Wieferich prime larger than 4×1012, andHis any abelian group for whichΣGhas the same periodic point data asΣH, thenGis isomorphic toH. This result may be viewed as an example of the “rigidity” properties of higher-dimensional Markov shifts with zero entropy.