Orbit-counting for nilpotent group shifts

Details Orbit-counting for nilpotent group shifts Orbit-counting for nilpotent group shifts

2007-06-25

arxiv.org/abs/0706.3630v2

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

Mathematics

Dynamical Systems

Scientific paper

10.1090/S0002-9939-08-09649-4

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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