Orbit-counting for nilpotent group shifts

Richard Miles, Thomas Ward

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1499-1507
Number of pages9
JournalProceedings of the American Mathematical Society
Issue number04
Publication statusPublished - 2008

Cite this