Orbit-counting for nilpotent group shifts

Miles, Richard; Ward, Thomas

doi:10.1090/S0002-9939-08-09649-4

Orbit-counting for nilpotent group shifts
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by Richard Miles and Thomas Ward PDF

Proc. Amer. Math. Soc. 137 (2009), 1499-1507 Request permission

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 \[ \sum _{\vert \tau \vert \le N}\frac {1}{e^{h\vert \tau \vert }}\sim CN^{\alpha }(\log N)^{\beta } \] 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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