Characterizations of periods of multidimensional shifts

Abstract

We show that the sets of periods of multidimensional shifts of finite type (SFTs) are exactly the sets of integers of the complexity class $\NE$. We also show that the functions counting their number are the functions of #E. We also give characterizations of some other notions of periodicity. We finish the paper by giving some characterizations for sofic and effective subshifts.