Periodic points classify a family of Markov shifts

Ledrappier introduced the following type of space of doubly indexed sequences over a finite abelian group G, The group acts naturally on the space XG via left and upward shifts. We show that the periodic point data of XG 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 and Teichmüller systems. Our approach unifies Wardʹs 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.