The number of configurations in the full shift with a given least period

TL;DR

This paper investigates the count of configurations with a specific least period in full shifts over groups, providing a formula involving the M"obius function and classifying cases with few orbits.

Contribution

It introduces a method to compute the number of configurations with a given least period using the M"obius function for finitely generated groups and classifies scenarios with limited orbits when the subgroup is normal.

Findings

Number of configurations computed via M"obius function for finite index subgroups.

Classification of cases with at most 10 orbits for normal subgroups.

Provides a formula linking subgroup structure to configuration counts.

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} $H \leq G$ 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 M\"obius 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 $10$.