R. Miles
Orbit-counting for nilpotent group shifts
Miles, R.; Ward, T.
Authors
T. Ward
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.
Citation
Miles, R., & Ward, T. (2009). Orbit-counting for nilpotent group shifts. Proceedings of the American Mathematical Society, 137(04), 1499-1507. https://doi.org/10.1090/s0002-9939-08-09649-4
Journal Article Type | Article |
---|---|
Publication Date | Jan 1, 2009 |
Deposit Date | Oct 12, 2012 |
Publicly Available Date | Dec 14, 2012 |
Journal | Proceedings of the American Mathematical Society |
Print ISSN | 0002-9939 |
Electronic ISSN | 1088-6826 |
Publisher | American Mathematical Society |
Peer Reviewed | Peer Reviewed |
Volume | 137 |
Issue | 04 |
Pages | 1499-1507 |
DOI | https://doi.org/10.1090/s0002-9939-08-09649-4 |
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Copyright Statement
First published in Transactions of the American Mathematical Society in 2009, volume 137 published by the American Mathematical Society. © Copyright 2009 American Mathematical Society.
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