Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. S. Agmon, Complex variable Tauberians, Trans. Amer. Math. Soc. 74 (1953), 444-481. MR 0054079 (14,869a)
  • 2. V. Chothi, G. Everest, and T. Ward, $ S$-integer dynamical systems: Periodic points, J. Reine Angew. Math. 489 (1997), 99-132. MR 1461206 (99b:11089)
  • 3. G. Everest, R. Miles, S. Stevens, and T. Ward, Orbit-counting in non-hyperbolic dynamical systems, J. Reine Angew. Math. 608 (2007), 155-182. MR 2339472 (2008k:37042)
  • 4. G. Everest, V. Stangoe, and T. Ward, Orbit counting with an isometric direction, Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, pp. 293-302. MR 2180241 (2006k:37046)
  • 5. G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964. MR 0185094 (32:2564)
  • 6. E. Hlawka, Discrepancy and uniform distribution of sequences, Compositio Math. 16 (1964), 83-91 (1964). MR 0174544 (30:4745)
  • 7. D. A. Lind and T. Ward, Automorphisms of solenoids and $ p$-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411-419. MR 961739 (90a:28031)
  • 8. R. Miles, Zeta functions for elements of entropy rank-one actions, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 567-582. MR 2308145 (2008h:37008)
  • 9. -, Periodic points of endomorphisms on solenoids and related groups, Bull. Lond. Math. Soc. 40 (2008), no. 4, 696-704. MR 2441142
  • 10. J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136. MR 0047262 (13,850e)
  • 11. W. Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions, Israel J. Math. 45 (1983), no. 1, 41-52. MR 710244 (85c:58089)
  • 12. W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2) 118 (1983), no. 3, 573-591. MR 727704 (85i:58105)
  • 13. -, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque (1990), no. 187-188, 268. MR 1085356 (92f:58141)
  • 14. D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc. 49 (2002), no. 8, 887-895. MR 1920859 (2003d:37026)
  • 15. V. Stangoe, Orbit counting far from hyperbolicity, Ph.D. thesis, University of East Anglia, 2004.
  • 16. R. P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997, with a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original. MR 1442260 (98a:05001)
  • 17. S. Waddington, The prime orbit theorem for quasihyperbolic toral automorphisms, Monatsh. Math. 112 (1991), no. 3, 235-248. MR 1139101 (92k:58219)
  • 18. T. Ward, Almost all $ S$-integer dynamical systems have many periodic points, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 471-486. MR 1619569 (99k:58152)
  • 19. -, Dynamical zeta functions for typical extensions of full shifts, Finite Fields Appl. 5 (1999), no. 3, 232-239. MR 1702897 (2000m:11067)
  • 20. H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313-352. MR 1511862

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37C30, 26E30, 12J25

Retrieve articles in all journals with MSC (2000): 37C30, 26E30, 12J25


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.

American Mathematical Society