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Orbit-counting for nilpotent group shifts. (English) Zbl 1160.22005
Summary: 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 \[ \sum_{|\tau|\leqslant N}\frac{1}{e^{h|\tau|}}\sim CN^{\alpha}(\log N)^{\beta} \] where \( |\tau|\) 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.

MSC:
37A15 General groups of measure-preserving transformations and dynamical systems
22D40 Ergodic theory on groups
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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