Dirichlet series for finite combinatorial rank dynamics

Authors:
G. Everest, R. Miles, S. Stevens and T. Ward

Journal:
Trans. Amer. Math. Soc. **362** (2010), 199-227

MSC (2000):
Primary 37C30; Secondary 26E30, 12J25

DOI:
https://doi.org/10.1090/S0002-9947-09-04962-9

Published electronically:
July 30, 2009

MathSciNet review:
2550149

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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 to be a rational function of exponential variables. 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.

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Additional Information

**G. Everest**

Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Email:
g.everest@uea.ac.uk

**R. Miles**

Affiliation:
Department of Mathematics, KTH-Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Email:
ricmiles@kth.se

**S. Stevens**

Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Email:
shaun.stevens@uea.ac.uk

**T. Ward**

Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Email:
t.ward@uea.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-09-04962-9

Received by editor(s):
July 25, 2007

Published electronically:
July 30, 2009

Additional Notes:
This research was supported by E.P.S.R.C. grant EP/C015754/1.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.