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Mertens’ theorem for toral automorphisms. (English) Zbl 1232.37017

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.

MSC:

37C35 Orbit growth in dynamical systems
11J72 Irrationality; linear independence over a field

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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
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