# Orbit-counting for nilpotent group shifts

Miles, R and Ward, T (2008) Orbit-counting for nilpotent group shifts. Proceedings of the American Mathematical Society, 137 (04). pp. 1499-1507. ISSN 0002-9939

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## 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.

Item Type: Article Faculty of Science > School of Mathematics Vishal Gautam 18 Mar 2011 14:48 26 Sep 2022 01:39 https://ueaeprints.uea.ac.uk/id/eprint/19750 10.1090/S0002-9939-08-09649-4