Dirichlet series for finite combinatorial rank dynamics

Everest, G.; Miles, R.; Stevens, S.; Ward, T.

doi:10.1090/s0002-9947-09-04962-9

Dirichlet series for finite combinatorial rank dynamics

Everest, G.; Miles, R.; Stevens, S.; Ward, T.

Authors

G. Everest

R. Miles

S. Stevens

T. Ward

Abstract

We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.

Citation

Everest, G., Miles, R., Stevens, S., & Ward, T. (2010). Dirichlet series for finite combinatorial rank dynamics. Transactions of the American Mathematical Society, 362(01), 199-227. https://doi.org/10.1090/s0002-9947-09-04962-9

Journal Article Type	Article
Publication Date	Jan 1, 2010
Deposit Date	Oct 12, 2012
Publicly Available Date	Dec 14, 2012
Journal	Transactions of the American Mathematical Society
Print ISSN	0002-9947
Electronic ISSN	1088-6850
Publisher	American Mathematical Society
Peer Reviewed	Peer Reviewed
Volume	362
Issue	01
Pages	199-227
DOI	https://doi.org/10.1090/s0002-9947-09-04962-9

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Copyright Statement
First published in Transactions of the American Mathematical Society in 2010, volume 362 published by the American Mathematical Society. © Copyright 2010 American Mathematical Society.