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 \emph{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 $\text{Aut}(A^G)$-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 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 $10$.

8 pages

References in corpus (2)

• Cellular Automata and Finite Groups
• On two Möbius function for a finite non-solvable group

Cited by in corpus (1)

• The relative rank of the endomorphism monoid of a finite $G$-set