Periodic points classify a family of Markov shifts

Abstract

Ledrappier introduced the following type of space of doubly indexed sequences over a finite abelian group G , X G ≔ ( x s , t ) ∈ G Z 2 | x s , t + 1 = x s , t + x s + 1 , t for all s , t ∈ Z . The group Z 2 acts naturally on the space X G via left and upward shifts. We show that the periodic point data of X G determine the group G up to isomorphism. This is extending work of Ward, using a new way to calculate periodic point numbers based on the study of polynomials over Z / p n / Z and Teichmüller systems. Our approach unifies Ward's treatment of the two known Wieferich primes with that of all other primes and settles the cases of arbitrary Wieferich primes and the prime two.