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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Dirichlet series for finite combinatorial rank dynamics
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by G. Everest, R. Miles, S. Stevens and T. Ward PDF
Trans. Amer. Math. Soc. 362 (2010), 199-227 Request permission

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
  • MR Author ID: 678092
  • Email: shaun.stevens@uea.ac.uk
  • T. Ward
  • Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
  • MR Author ID: 180610
  • Email: t.ward@uea.ac.uk
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2550149