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Orbit-counting for nilpotent group shifts

Miles, R.; Ward, T.

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Authors

R. Miles

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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