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Orbit-counting for nilpotent group shifts
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2009 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 137, no 4, 1499-1507 p.Article in journal (Refereed) Published
Abstract [en]

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full G-shift for a. nitely-generated torsion-free nilpotent group G. Using bounds for the Mobius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape Sigma vertical bar(tau vertical bar <= N)1/e(h)vertical bar tau vertical bar similar to C N-alpha(log N)(beta) where vertical bar tau vertical bar 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 noncyclic groups there is no single asymptotic in terms of elementary functions.

Place, publisher, year, edition, pages
2009. Vol. 137, no 4, 1499-1507 p.
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URN: urn:nbn:se:kth:diva-32517DOI: 10.1090/S0002-9939-08-09649-4ISI: 000261968800041ScopusID: 2-s2.0-77950575924OAI: diva2:410937
QC 20110415Available from: 2011-04-15 Created: 2011-04-15 Last updated: 2011-04-15Bibliographically approved

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Miles, Richard
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