The Number of Configurations in the Full Shift with a Given Least Period

Abstract

For any group G and any set A, consider the shift action of G on the full shift \(A^G\). A configuration \(x \in A^G\) has least period \(H \le G\) if the stabiliser of x is precisely H. Among other things, the number of such configurations is interesting as it provides an upper bound for the size of the corresponding \(\mathrm {Aut}(A^G)\)-orbit. In this paper, we show that if G is finitely generated and H is of finite index, then the number of configurations in \(A^G\) with least period H may be computed by using the Möbius function of the lattice of subgroups of finite index in G. Moreover, when H is a normal subgroup, we classify all situations such that the number of G-orbits with least period H is at most 10.