Orbit-counting for nilpotent group shifts

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Miles, Richard and Ward, Thomas (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 \ School:	Faculty of Science > School of Mathematics
Related URLs:	http://arxiv.org/abs/0706.3630 http://www.zentralblatt-math.org/zmath/e...
Depositing User:	Vishal Gautam
Date Deposited:	18 Mar 2011 14:48
Last Modified:	24 Jan 2023 10:30
URI:	https://ueaeprints.uea.ac.uk/id/eprint/19750
DOI:	10.1090/S0002-9939-08-09649-4

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