Orbit-counting for nilpotent group shifts

Abstract

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{ö}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum_{|τ|\le N}\frac{1}{e^{h|τ|}}\sim CN^α(\log N)^β \] where $|τ|$ is the cardinality of the finite orbit $τ$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.