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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Orbit-counting for nilpotent group shifts


Authors: Richard Miles and Thomas Ward
Journal: Proc. Amer. Math. Soc. 137 (2009), 1499-1507
MSC (2000): Primary 22D40, 37A15, 37A35
Published electronically: October 23, 2008
MathSciNet review: 2465676
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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 Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape

$\displaystyle \sum_{\vert\tau\vert\le N}\frac{1}{e^{h\vert\tau\vert}}\sim CN^{\alpha}(\log N)^{\beta} $

where  $ \vert\tau\vert$ is the cardinality of the finite orbit $ \tau$ and $ h$ denotes the topological entropy. 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.


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Additional Information

Richard Miles
Affiliation: School of Mathematics, KTH, SE-100 44, Stockholm, Sweden

Thomas Ward
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09649-4
PII: S 0002-9939(08)09649-4
Received by editor(s): June 25, 2007
Received by editor(s) in revised form: August 22, 2007, and May 5, 2008
Published electronically: October 23, 2008
Additional Notes: We thank Johannes Siemons and Shaun Stevens for their suggestions. This research was supported by E.P.S.R.C. grant EP/C015754/1.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.