# Orbit-counting for nilpotent group shifts

R Miles, T Ward

Research output: Contribution to journalArticle

11 Citations (Scopus)

## 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{\"o}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape $\sum_{|\tau|\le N}\frac{1}{e^{h|\tau|}}\sim CN^{\alpha} (\log N)^{\beta}$ where $|\tau|$ is the cardinality of the finite orbit $\tau$. 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.
Original language English 1499-1507 9 Proceedings of the American Mathematical Society 137 04 https://doi.org/10.1090/S0002-9939-08-09649-4 Published - 2008