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Orbit Growth of Sofic Shifts and Periodic-Finite-Type Shifts

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Abstract

A sofic shift is a shift space consisting of bi-infinite labels of paths from a labelled graph. Being a dynamical system, the distribution of its closed orbits may indicate the complexity of the shift. For this purpose, prime orbit and Mertens’ orbit counting functions are introduced as a way to describe the growth of the closed orbits. The asymptotic behaviours of these counting functions can be implied from the analyticity of the Artin–Mazur zeta function of the shift. Its zeta function is expressed implicitly in terms of several signed subset matrices. In this paper, we will prove the asymptotic behaviours of the counting functions for sofic shifts via their zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof simply uses some well-known facts about sofic shifts, especially on the minimal right-resolving presentations. Furthermore, we will demonstrate this result by revisiting the case for periodic-finite-type shifts, which are a particular type of sofic shifts. At the end, we will briefly discuss the application of our finding towards the finite group and homogeneous extensions of a sofic shift.

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References

  1. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)

    Book  Google Scholar 

  2. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–188, 1–255 (1990)

    Google Scholar 

  3. Artin, M., Mazur, B.: On periodic points. Ann. Math. 81(1), 82–99 (1965). https://doi.org/10.2307/1970384

    Article  MathSciNet  Google Scholar 

  4. Nordin, A., Noorani, M.S.M.: Orbit growth of periodic-finite-type shifts via Artin–Mazur zeta function. Mathematics 8(5), 685 (2020). https://doi.org/10.3390/math8050685

    Article  Google Scholar 

  5. Noorani, M.S.M.: Mertens theorem and closed orbits of ergodic toral automorphisms. Bull. Malays. Math. Sci. Soc. 22, 127–133 (1999)

    MathSciNet  Google Scholar 

  6. Waddington, S.: The prime orbit theorem for quasihyperbolic toral automorphisms. Monatshefte für Mathematik 112, 235–248 (1991). https://doi.org/10.1007/BF01297343

    Article  MathSciNet  Google Scholar 

  7. Parry, W.: An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Isr. J. Math. 45(1), 41–52 (1983). https://doi.org/10.1007/BF02760669

    Article  MathSciNet  Google Scholar 

  8. Nordin, A., Noorani, M.S.M., Dzul-Kifli, S.C.: Orbit growth of Dyck and Motzkin shifts via Artin–Mazur zeta function. Dyn. Syst. 35(4), 655–667 (2020). https://doi.org/10.1080/14689367.2020.1770201

    Article  MathSciNet  Google Scholar 

  9. Nordin, A., Noorani, M.S.M.: Orbit growth of shift spaces induced by bouquet graphs and Dyck shifts. Mathematics 9(11), 1–10 (2021). https://doi.org/10.3390/math9111268

    Article  Google Scholar 

  10. Flatto, L., Lagarias, J.C., Poonen, B.: The zeta function of the beta transformation. Ergod. Theory Dyn. Syst. 14(2), 237–266 (1994). https://doi.org/10.1017/S0143385700007860

    Article  MathSciNet  Google Scholar 

  11. Nguema Ndong, F.: Zeta function and negative beta-shifts. Monatshefte für Mathematik 188, 717–751 (2019). https://doi.org/10.1007/s00605-019-01271-z

    Article  MathSciNet  Google Scholar 

  12. Buzzi, J.: Subshifts of quasi-finite type. Inventiones mathematicae 159, 369–406 (2005). https://doi.org/10.1007/s00222-004-0392-1

    Article  MathSciNet  Google Scholar 

  13. Everest, G., Miles, R., Stevens, S., Ward, T.: Dirichlet series for finite combinatorial rank dynamics. Trans. Am. Math. Soc. 362(1), 199–227 (2009). https://doi.org/10.1090/S0002-9947-09-04962-9

    Article  MathSciNet  Google Scholar 

  14. Pakapongpun, A., Ward, T.: Functorial orbit counting. J. Integer Seq. 12, 1–20 (2009)

    MathSciNet  Google Scholar 

  15. Akhatkulov, S., Noorani, M.S.M., Akhadkulov, H.: An analogue of the prime number, Mertens’ and Meissel’s theorems for closed orbits of the Dyck shift. In: AIP Conference Proceedings, vol. 1830, pp. 1–9 (2017). https://doi.org/10.1063/1.4980971

  16. Alsharari, F., Noorani, M.S.M., Akhadkulov, H.: Estimates on the number of orbits of the Dyck shift. J. Inequalities Appl. 2015, 1–12 (2015). https://doi.org/10.1186/s13660-015-0899-6

    Article  MathSciNet  Google Scholar 

  17. Alsharari, F., Noorani, M.S.M., Akhadkulov, H.: Analogues of the prime number theorem and Mertens’ theorem for closed orbits of the Motzkin shift. Bull. Malays. Math. Sci. Soc. 40, 307–319 (2017). https://doi.org/10.1007/s40840-015-0144-y

    Article  MathSciNet  Google Scholar 

  18. Miles, R., Ward, T.: Orbit-counting for nilpotent group shifts. Proc. Am. Math. Soc. 137(4), 1499–1507 (2008). https://doi.org/10.1090/S0002-9939-08-09649-4

    Article  MathSciNet  Google Scholar 

  19. Miles, R.: Orbit growth for algebraic flip systems. Ergodic Theory Dyn. Syst. 35(8), 2613–2631 (2015). https://doi.org/10.1017/etds.2014.38

    Article  MathSciNet  Google Scholar 

  20. Nordin, A., Noorani, M.S.M.: Counting finite orbits for the flip systems of shifts of finite type. Discret. Contin. Dyn. Syst. Ser. A 41(10), 4515–4529 (2021). https://doi.org/10.3934/dcds.2021046

    Article  MathSciNet  Google Scholar 

  21. Nordin, A., Noorani, M.S.M., Dzul-Kifli, S.C.: Counting closed orbits in discrete dynamical systems. In: Mohd, M., Abdul Rahman, N., Abd Hamid, N., Mohd Yatim, Y. (eds.) Dynamical Systems, Bifurcation Analysis and Applications, pp. 147–171. Springer, Singapore (2018). https://doi.org/10.1007/978-981-32-9832-3_9

  22. Ward, T.: Ergodic theory: interactions with combinatorics and number theory. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 3040–3053. Springer, New York (2009). https://doi.org/10.1007/978-0-387-30440-3

  23. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995). https://doi.org/10.1017/CBO9780511626302

  24. Béal, M.-P., Crochemore, M., Moision, B.E., Siegel, P.H.: Periodic-finite-type shift spaces. IEEE Trans. Inf. Theory 57(6), 3677–3691 (2011). https://doi.org/10.1109/TIT.2011.2143910

    Article  MathSciNet  Google Scholar 

  25. Manada, A., Kashyap, N.: On the zeta function of a periodic-finite-type shift. IEICE Trans. Fundamentals Electron. Commun. Comput. Sci. E96.A(6), 1024–1031 (2013). https://doi.org/10.1587/transfun.E96.A.1024

  26. Nordin, A., Noorani, M.S.M.: A short note on the orbit growth of sofic shifts. arXiv arXiv:2202.03075 (2022)

  27. Mohamed, M., Noorani, M.S.M.: Teorem Mertens bagi orbit-orbit tertutup subanjakan mengikut kelas Frobenius. Matematika 15(2), 119–127 (1999)

    Google Scholar 

  28. Noorani, M.S.M., Parry, W.: A Chebotarev Theorem for finite homogeneous extensions of shifts. Bol. Soc. Brasil. Mat. 23(1–2), 137–151 (1992). https://doi.org/10.1007/BF02584816

    Article  MathSciNet  Google Scholar 

  29. Béal, M.-P., Blockelet, M., Dima, C.: Sofic-Dyck shifts. Theoret. Comput. Sci. 609(1), 226–244 (2016). https://doi.org/10.1016/j.tcs.2015.09.027

    Article  MathSciNet  Google Scholar 

  30. Krieger, W., Matsumoto, K.: Zeta functions and topological entropy of the Markov–Dyck shifts. Münster J. Math. 4, 171–184 (2011)

    MathSciNet  Google Scholar 

  31. Inoue, K., Krieger, W.: Subshifts from sofic shifts and Dyck shifts, zeta functions and topological entropy. arXiv 1–19 (2010). arXiv:1001.1839

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Acknowledgements

The first author would like to thank Universiti Sains Malaysia for the financial support to conduct research in the university under Post-Doctoral Fellowship Scheme. We are deeply thankful to the referees for constructive comments and suggestions which help to improve the quality of this paper.

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Author notes

  1. Mohd Salmi Md Noorani and Mohd Hafiz Mohd contributed equally to this work.

Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang, 11800, Penang, Malaysia

    Azmeer Nordin & Mohd Hafiz Mohd

  2. Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi, 43600, Selangor, Malaysia

    Mohd Salmi Md Noorani

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  1. Azmeer Nordin
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  2. Mohd Salmi Md Noorani
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  3. Mohd Hafiz Mohd
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Correspondence to Azmeer Nordin.

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Nordin, A., Noorani, M.S.M. & Mohd, M.H. Orbit Growth of Sofic Shifts and Periodic-Finite-Type Shifts. Qual. Theory Dyn. Syst. 23, 188 (2024). https://doi.org/10.1007/s12346-024-01055-3

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