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

## 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 |
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Faculty \ School: | Faculty of Science > School of Mathematics |

Related URLs: | |

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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