Characterizations of periods of multidimensional shifts

We show that the sets of periods of multidimensional shifts of finite type (SFTs) are exactly the sets of integers of the complexity class $\text{NE}$. 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.