Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Orbit-counting for nilpotent group shifts

Author(s): Richard Miles; Thomas Ward
Journal: Proc. Amer. Math. Soc. 137 (2009), 1499-1507.
MSC (2000): Primary 22D40, 37A15, 37A35
Posted: October 23, 2008
MathSciNet review: 2465676
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full -shift for a finitely-generated torsion-free nilpotent group . 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

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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