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Orbit-counting for nilpotent group shifts.

Miles, R. and Ward, T. (2009) 'Orbit-counting for nilpotent group shifts.', Proceedings of the American Mathematical Society., 137 (04). pp. 1499-1507.


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.

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Publisher 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.
Record Created:12 Oct 2012 10:05
Last Modified:14 Dec 2012 13:59

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