Jaidee, Sawian; Stevens, Shaun; Ward, Thomas Mertens’ theorem for toral automorphisms. (English) Zbl 1232.37017 Proc. Am. Math. Soc. 139, No. 5, 1819-1824 (2011). The authors prove an interesting result (Theorem 1) regarding the ergodic toral automorphisms with error term \(O(N^{-1})\). The influence of resonances among the eigenvalues of unit modulus is examined. The properties are illustrated giving some significant examples. Reviewer: Nicolae-Adrian Secelean (Sibiu) Cited in 6 Documents MSC: 37C35 Orbit growth in dynamical systems 11J72 Irrationality; linear independence over a field Keywords:ergodic map; quasihyperbolic map; toral automorphism; Merten’s theorem Software:OEIS PDFBibTeX XMLCite \textit{S. Jaidee} et al., Proc. Am. Math. Soc. 139, No. 5, 1819--1824 (2011; Zbl 1232.37017) Full Text: DOI arXiv References: [1] V. Chothi, G. Everest, and T. Ward, \?-integer dynamical systems: periodic points, J. Reine Angew. Math. 489 (1997), 99 – 132. · Zbl 0879.58037 [2] 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 [3] Graham Everest and Thomas Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer-Verlag London, Ltd., London, 1999. · Zbl 0919.11064 [4] Mohd. Salmi Md. Noorani, Mertens theorem and closed orbits of ergodic toral automorphisms, Bull. Malaysian Math. Soc. (2) 22 (1999), no. 2, 127 – 133. · Zbl 1142.37314 [5] Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, J. Integer Seq. 12 (2009), no. 2, Article 09.2.4, 20. · Zbl 1254.37020 [6] 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 [7] N. J. A. Sloane. An on-line version of the encyclopedia of integer sequences. Electron. J. Combin. 1: Feature 1, approx. 5 pp., 1994. www.research.att.com/ñjas/sequences/. · Zbl 0815.11001 [8] Simon Waddington, The prime orbit theorem for quasihyperbolic toral automorphisms, Monatsh. Math. 112 (1991), no. 3, 235 – 248. · Zbl 0737.28008 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.