Miles, Richard; Ward, Thomas The dynamical zeta function for commuting automorphisms of zero-dimensional groups. (English) Zbl 1400.37027 Ergodic Theory Dyn. Syst. 38, No. 4, 1564-1587 (2018). D. A. Lind [Lond. Math. Soc. Lect. Note Ser. 228, 433–450 (1996; Zbl 0881.58052)] introduced a dynamical zeta function for a \({\mathbb{Z}}^d\)-action \(A\) defined by commuting homeomorphisms of a compact metric space. The authors study the same object when \(A\) is generated by continuous automorphisms of a compact abelian zero-dimensional group. They address Lind’s conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. Reviewer: Michael L. Blank (Moskva) Cited in 1 Document MSC: 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 22E10 General properties and structure of complex Lie groups 37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) Keywords:\({\mathbb{Z}}^d\)-action; commuting homeomorphisms; Lind’s conjecture; dynamical zeta function Citations:Zbl 0881.58052 PDFBibTeX XMLCite \textit{R. Miles} and \textit{T. Ward}, Ergodic Theory Dyn. Syst. 38, No. 4, 1564--1587 (2018; Zbl 1400.37027) Full Text: DOI arXiv Link References: [1] M.Artin and B.Mazur. On periodic points. Ann. of Math. (2)81 (1965), 82-99.10.2307/1970384 · Zbl 0127.13401 [2] J.Bell, R.Miles and T.Ward. Towards a Pólya-Carlson dichotomy for algebraic dynamics. Indag. Math. (N.S.)25(4) (2014), 652-668.10.1016/j.indag.2014.04.005 · Zbl 1319.37012 [3] F.Carlson. Über ganzwertige Funktionen. Math. Z.11(1-2) (1921), 1-23.10.1007/BF01203188 · JFM 48.0387.01 [4] Y.Choie, N.Lichiardopol, P.Moree and P.Solé. On Robin’s criterion for the Riemann hypothesis. J. Théor. Nombres Bordeaux19(2) (2007), 357-372.10.5802/jtnb.591 · Zbl 1163.11059 [5] V.Chothi, G.Everest and T.Ward. S-integer dynamical systems: periodic points. J. Reine Angew. Math.489 (1997), 99-132. · Zbl 0879.58037 [6] P.Corvaja and U.Zannier. A lower bound for the height of a rational function at S-unit points. Monatsh. Math.144(3) (2005), 203-224.10.1007/s00605-004-0273-0 · Zbl 1086.11035 [7] P.Corvaja and U.Zannier. Greatest common divisors of u - 1, v - 1 in positive characteristic and rational points on curves over finite fields. J. Eur. Math. Soc. (JEMS)15(5) (2013), 1927-1942.10.4171/JEMS/409 · Zbl 1325.11060 [8] M.Einsiedler and D.Lind. Algebraic ℤ^d-actions of entropy rank one. Trans. Amer. Math. Soc.356(5) (2004), 1799-1831.10.1090/S0002-9947-04-03554-8 · Zbl 1033.37002 [9] G.Everest, A.van der Poorten, I.Shparlinski and T.Ward. Recurrence Sequences(Mathematical Surveys and Monographs, 104). American Mathematical Society, Providence, RI, 2003.10.1090/surv/104 · Zbl 1033.11006 [10] G.Everest and T.Ward. Heights of polynomials and entropy in algebraic dynamics. Universitext. Springer, London, 1999. · Zbl 0919.11064 [11] T. H.Gronwall. Some asymptotic expressions in the theory of numbers. Trans. Amer. Math. Soc.14(1) (1913), 113-122.10.1090/S0002-9947-1913-1500940-6 · JFM 44.0236.03 [12] B.Kitchens and K.Schmidt. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys.9(4) (1989), 691-735.10.1017/S0143385700005290 · Zbl 0709.54023 [13] P.Kurlberg and C.Pomerance. On the periods of the linear congruential and power generators. Acta Arith.119(2) (2005), 149-169.10.4064/aa119-2-2 · Zbl 1080.11059 [14] F.Ledrappier. Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris A-B287(7) (1978), A561-A563. · Zbl 0387.60084 [15] R.Lidl and H.Niederreiter. Introduction to Finite Fields and their Applications, 1st edn. Cambridge University Press, Cambridge, 1994.10.1017/CBO9781139172769 · Zbl 0820.11072 [16] D. A.Lind. A zeta function for ℤ^d-actions. Ergodic Theory of ℤ^d-actions (Warwick, 1993-1994)(London Mathematical Society Lecture Note Series, 228). Cambridge University Press, Cambridge, 1996, pp. 433-450. · Zbl 0881.58052 [17] D.Lind, K.Schmidt and T.Ward. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math.101(3) (1990), 593-629.10.1007/BF01231517 · Zbl 0774.22002 [18] H.Matsumura. Commutative Ring Theory, 2nd edn.(Cambridge Studies in Advanced Mathematics, 8). Cambridge University Press, Cambridge, 1989. · Zbl 0666.13002 [19] R.Miles. Zeta functions for elements of entropy rank-one actions. Ergod. Th. & Dyn. Sys.27(2) (2007), 567-582.10.1017/S0143385706000794 · Zbl 1110.37013 [20] R.Miles. Periodic points of endomorphisms on solenoids and related groups. Bull. Lond. Math. Soc.40(4) (2008), 696-704.10.1112/blms/bdn052 · Zbl 1147.37012 [21] R.Miles. Synchronization points and associated dynamical invariants. Trans. Amer. Math. Soc.365(10) (2013), 5503-5524.10.1090/S0002-9947-2013-05829-1 · Zbl 1350.37023 [22] R.Miles. A natural boundary for the dynamical zeta function for commuting group automorphisms. Proc. Amer. Math. Soc.143(7) (2015), 2927-2933.10.1090/S0002-9939-2015-12515-4 · Zbl 1364.37019 [23] G.Pólya. Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe. Math. Ann.99(1) (1928), 687-706.10.1007/BF01459120 · JFM 54.0340.07 [24] K.Schmidt. Dynamical Systems of Algebraic Origin(Progress in Mathematics, 128). Birkhäuser, Basel, 1995. · Zbl 0833.28001 [25] S. L.Segal. Nine Introductions in Complex Analysis(North-Holland Mathematics Studies, 208), revised edn. Elsevier Science B.V., Amsterdam, 2008. · Zbl 1233.30001 [26] J. H.Silverman. Common divisors of a^n - 1 and b^n - 1 over function fields. New York J. Math.10 (2004), 37-43. · Zbl 1120.11045 [27] T.Ward. Dynamical zeta functions for typical extensions of full shifts. Finite Fields Appl.5(3) (1999), 232-239.10.1006/ffta.1999.0250 · Zbl 0956.37016 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.