Miles, Richard; Ward, Thomas B. Orbit-counting for nilpotent group shifts. (English) Zbl 1160.22005 Proc. Am. Math. Soc. 137, No. 4, 1499-1507 (2009). Summary: We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full \( G\)-shift for a finitely-generated torsion-free nilpotent group \( G\). Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum_{|\tau|\leqslant N}\frac{1}{e^{h|\tau|}}\sim CN^{\alpha}(\log N)^{\beta} \] where \( |\tau|\) is the cardinality of the finite orbit \( \tau\) and \( h\) denotes the topological entropy. For the usual orbit-counting function we find upper and lower bounds, together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions. Cited in 2 ReviewsCited in 8 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 22D40 Ergodic theory on groups 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) PDFBibTeX XMLCite \textit{R. Miles} and \textit{T. B. Ward}, Proc. Am. Math. Soc. 137, No. 4, 1499--1507 (2009; Zbl 1160.22005) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Denominator of the rational coefficient in the main term in the dynamical analog of Mertens’s theorem for a full n-dimensional shift, n >= 2. Numerator of the rational coefficient in the main term in the dynamical analog of Mertens’s theorem for a full n-dimensional shift, n >= 12 (it is 1 for 2 <= n <= 11). References: [1] Henry H. Crapo, Möbius inversion in lattices, Arch. Math. (Basel) 19 (1968), 595 – 607 (1969). · Zbl 0208.29303 [2] Marcus du Sautoy and Fritz Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), no. 3, 793 – 833. · Zbl 1006.11051 [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. · Zbl 1137.37006 [4] F. J. Grunewald, D. Segal, and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), no. 1, 185 – 223. · Zbl 0651.20040 [5] Charles Kratzer and Jacques Thévenaz, Fonction de Möbius d’un groupe fini et anneau de Burnside, Comment. Math. Helv. 59 (1984), no. 3, 425 – 438 (French). · Zbl 0592.20004 [6] François Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A561 – A563 (French, with English summary). · Zbl 0387.60084 [7] Douglas Lind, Klaus Schmidt, and Tom Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), no. 3, 593 – 629. · Zbl 0774.22002 [8] D. A. Lind, A zeta function for \?^{\?}-actions, Ergodic theory of \?^{\?} actions (Warwick, 1993 – 1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 433 – 450. · Zbl 0881.58052 [9] Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. · Zbl 1071.20033 [10] Grigoriy A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows; Translated from the Russian by Valentina Vladimirovna Szulikowska. · Zbl 1140.37010 [11] 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. · Zbl 0537.58038 [12] Richard Sharp, An analogue of Mertens’ theorem for closed orbits of Axiom A flows, Bol. Soc. Brasil. Mat. (N.S.) 21 (1991), no. 2, 205 – 229. · Zbl 0761.58041 [13] G. C. Smith, Zeta functions of torsion free finitely generated nilpotent groups, Ph.D. thesis, Manchester (UMIST), 1983. [14] Richard 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. · Zbl 0889.05001 [15] T. B. Ward, Periodic points for expansive actions of \?^{\?} on compact abelian groups, Bull. London Math. Soc. 24 (1992), no. 4, 317 – 324. · Zbl 0725.22003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.