×

Estimates on the number of orbits of the Dyck shift. (English) Zbl 1353.37046

Summary: In this paper, we get crucial estimates of fundamental sums that involve the number of closed orbits of the Dyck shift. These estimates are given as the prime orbit theorem, Mertens’ orbit theorem, Meissel’s orbit theorem and Dirichlet series. Different and more direct methods are used in the proofs without any complicated theoretical discussions.

MSC:

37C35 Orbit growth in dynamical systems
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lindqvist, P, Peetre, J: On a number theoretic sum considered by Meissel. Nieuw Arch. Wiskd. 15, 175-179 (1997) · Zbl 0910.11035
[2] Patterson, SJ: An Introduction to the Theory of the Riemann Zeta-Function. Cambridge Studies in Advanced Mathematics, vol. 14. Cambridge University Press, Cambridge (1988) · Zbl 0641.10029 · doi:10.1017/CBO9780511623707
[3] Hardy, GH, Wright, EM: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1938)
[4] Artin, M, Mazur, B: On periodic points. Ann. Math. 81(2), 82-99 (1965) · Zbl 0127.13401 · doi:10.2307/1970384
[5] Parry, W: An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Isr. J. Math. 45, 41-52 (1983) · Zbl 0552.28020 · doi:10.1007/BF02760669
[6] Parry, W, Pollicott, M: An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. Math. 118, 573-591 (1983) · Zbl 0537.58038 · doi:10.2307/2006982
[7] Noorani, MSM: Counting closed orbits of hyperbolic diffeomorphisms. Results Math. 50, 241-257 (2007) · Zbl 1145.37017 · doi:10.1007/s00025-007-0250-8
[8] Sharp, R: An analogue of Mertens’ theorem for closed orbits of Axiom A flows. Bol. Soc. Bras. Mat. 21, 205-229 (1991) · Zbl 0761.58041 · doi:10.1007/BF01237365
[9] Everest, D, Miles, R, Stevens, S, Ward, T: Orbit-counting in non-hyperbolic dynamical systems. J. Reine Angew. Math. 608, 155-182 (2007) · Zbl 1137.37006
[10] Noorani, MSM: Mertens’ theorem and closed orbits of ergodic toral automorphisms. Bull. Malays. Math. Sci. Soc. (2) 22, 127-133 (1999) · Zbl 1142.37314
[11] Waddington, S: The prime orbit theorem for quasihyperbolic toral automorphisms. Monatshefte Math. 112, 235-248 (1991) · Zbl 0737.28008 · doi:10.1007/BF01297343
[12] Jaidee, S, Stevens, S, Ward, T: Mertens’ theorem for toral automorphisms. Proc. Am. Math. Soc. 139, 1819-1824 (2011) · Zbl 1232.37017 · doi:10.1090/S0002-9939-2010-10632-9
[13] Alsharari, F, Noorani, MSM, Akhadkulov, H: Counting closed orbits for the Dyck shift. Abstr. Appl. Anal. 2014, Article ID 304798 (2014) · Zbl 1449.37017 · doi:10.1155/2014/304798
[14] Marcus, B, Lind, D: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995) · Zbl 1106.37301
[15] Krieger, W: On the uniqueness of equilibrium state. Math. Syst. Theory 8, 97-104 (1974) · Zbl 0302.28011 · doi:10.1007/BF01762180
[16] Hamachi, T, Inoue, K: Embedding shifts of finite type into the Dyck shift. Monatshefte Math. 145, 107-129 (2005) · Zbl 1181.37009 · doi:10.1007/s00605-004-0297-5
[17] Gerhard, K: Circular codes, loop counting, and zeta-functions. J. Comb. Theory 56, 75-83 (1991) · Zbl 0718.94013 · doi:10.1016/0097-3165(91)90023-A
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.