Dirichlet series for finite combinatorial rank dynamics
Authors:
G. Everest, R. Miles, S. Stevens and T. Ward
Journal:
Trans. Amer. Math. Soc. 362 (2010), 199227
MSC (2000):
Primary 37C30; Secondary 26E30, 12J25
Published electronically:
July 30, 2009
MathSciNet review:
2550149
Fulltext PDF
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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 orbitgrowth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbitgrowth asymptotics are found, all of which are polynomially bounded.
 1.
Shmuel
Agmon, Complex variable Tauberians,
Trans. Amer. Math. Soc. 74 (1953), 444–481. MR 0054079
(14,869a), 10.1090/S00029947195300540795
 2.
V.
Chothi, G.
Everest, and T.
Ward, 𝑆integer 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, Orbitcounting in nonhyperbolic dynamical systems, J.
Reine Angew. Math. 608 (2007), 155–182. MR 2339472
(2008k:37042), 10.1515/CRELLE.2007.056
 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), 10.1090/conm/385/07202
 5.
G.
H. Hardy and M.
Riesz, The general theory of Dirichlet’s series,
Cambridge Tracts in Mathematics and Mathematical Physics, No. 18,
StechertHafner, Inc., New York, 1964. MR 0185094
(32 #2564)
 6.
Edmund
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 𝑝adic entropy,
Ergodic Theory Dynam. Systems 8 (1988), no. 3,
411–419. MR
961739 (90a:28031), 10.1017/S0143385700004545
 8.
Richard
Miles, Zeta functions for elements of entropy rankone
actions, Ergodic Theory Dynam. Systems 27 (2007),
no. 2, 567–582. MR 2308145
(2008h:37008), 10.1017/S0143385706000794
 9.
Richard
Miles, Periodic points of endomorphisms on solenoids and related
groups, Bull. Lond. Math. Soc. 40 (2008), no. 4,
696–704. MR 2441142
(2009e:37015), 10.1112/blms/bdn052
 10.
John
C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 0047262
(13,850e), 10.1090/S00029904195209580X
 11.
William
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), 10.1007/BF02760669
 12.
William
Parry and Mark
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), 10.2307/2006982
 13.
William
Parry and Mark
Pollicott, Zeta functions and the periodic orbit structure of
hyperbolic dynamics, Astérisque 187188
(1990), 268 (English, with French summary). MR 1085356
(92f:58141)
 14.
David
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.
Richard
P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge
Studies in Advanced Mathematics, vol. 49, Cambridge University Press,
Cambridge, 1997. With a foreword by GianCarlo Rota; Corrected reprint of
the 1986 original. MR 1442260
(98a:05001)
 17.
Simon
Waddington, The prime orbit theorem for quasihyperbolic toral
automorphisms, Monatsh. Math. 112 (1991), no. 3,
235–248. MR 1139101
(92k:58219), 10.1007/BF01297343
 18.
T.
B. Ward, Almost all 𝑆integer dynamical systems have many
periodic points, Ergodic Theory Dynam. Systems 18
(1998), no. 2, 471–486. MR 1619569
(99k:58152), 10.1017/S0143385798113378
 19.
T.
Ward, Dynamical zeta functions for typical extensions of full
shifts, Finite Fields Appl. 5 (1999), no. 3,
232–239. MR 1702897
(2000m:11067), 10.1006/ffta.1999.0250
 20.
Hermann
Weyl, Über die Gleichverteilung von Zahlen mod. Eins,
Math. Ann. 77 (1916), no. 3, 313–352 (German).
MR
1511862, 10.1007/BF01475864
 1.
 S. Agmon, Complex variable Tauberians, Trans. Amer. Math. Soc. 74 (1953), 444481. MR 0054079 (14,869a)
 2.
 V. Chothi, G. Everest, and T. Ward, integer dynamical systems: Periodic points, J. Reine Angew. Math. 489 (1997), 99132. MR 1461206 (99b:11089)
 3.
 G. Everest, R. Miles, S. Stevens, and T. Ward, Orbitcounting in nonhyperbolic dynamical systems, J. Reine Angew. Math. 608 (2007), 155182. 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. 293302. 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, StechertHafner, Inc., New York, 1964. MR 0185094 (32:2564)
 6.
 E. Hlawka, Discrepancy and uniform distribution of sequences, Compositio Math. 16 (1964), 8391 (1964). MR 0174544 (30:4745)
 7.
 D. A. Lind and T. Ward, Automorphisms of solenoids and adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411419. MR 961739 (90a:28031)
 8.
 R. Miles, Zeta functions for elements of entropy rankone actions, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 567582. MR 2308145 (2008h:37008)
 9.
 , Periodic points of endomorphisms on solenoids and related groups, Bull. Lond. Math. Soc. 40 (2008), no. 4, 696704. MR 2441142
 10.
 J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116136. 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, 4152. 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, 573591. MR 727704 (85i:58105)
 13.
 , Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque (1990), no. 187188, 268. MR 1085356 (92f:58141)
 14.
 D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc. 49 (2002), no. 8, 887895. 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 GianCarlo 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, 235248. MR 1139101 (92k:58219)
 18.
 T. Ward, Almost all integer dynamical systems have many periodic points, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 471486. MR 1619569 (99k:58152)
 19.
 , Dynamical zeta functions for typical extensions of full shifts, Finite Fields Appl. 5 (1999), no. 3, 232239. MR 1702897 (2000m:11067)
 20.
 H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313352. MR 1511862
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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, KTHRoyal Institute of Technology, SE100 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:
http://dx.doi.org/10.1090/S0002994709049629
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.
