Ir al contenido

Documat

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

  • Azmeer Nordin [1] ; Mohd Salmi Md Noorani [2] ; Mohd Hafiz Mohd [1]
    1. [1] Universiti Sains Malaysia

      Universiti Sains Malaysia

      Malasia

    2. [2] National University of Malaysia

      National University of Malaysia

      Malasia

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
  • Idioma: inglés
  • DOI: 10.1007/s12346-024-01055-3
  • Enlaces
  • Resumen

    • 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 rightresolving 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.

  • Referencias bibliográficas
    • 1. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)
    • 2. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–188, 1–255 (1990)
    • 3. Artin, M., Mazur, B.: On periodic points. Ann. Math. 81(1), 82–99 (1965). https://doi.org/10.2307/ 1970384
    • 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
    • 5. Noorani, M.S.M.: Mertens theorem and closed orbits of ergodic toral automorphisms. Bull. Malays. Math. Sci. Soc. 22, 127–133 (1999)
    • 6. Waddington, S.: The prime orbit theorem for quasihyperbolic toral automorphisms. Monatshefte für Mathematik 112, 235–248 (1991). https://doi.org/10.1007/BF01297343
    • 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),...
    • 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),...
    • 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
    • 10. Flatto, L., Lagarias, J.C., Poonen, B.: The zeta function of the beta transformation. Ergod. Theory Dyn. Syst. 14(2), 237–266 (1994)....
    • 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
    • 12. Buzzi, J.: Subshifts of quasi-finite type. Inventiones mathematicae 159, 369–406 (2005). https://doi. org/10.1007/s00222-004-0392-1
    • 13. Everest, G., Miles, R., Stevens, S., Ward, T.: Dirichlet series for finite combinatorial rank dynamics. Trans. Am. Math. Soc. 362(1),...
    • 14. Pakapongpun, A., Ward, T.: Functorial orbit counting. J. Integer Seq. 12, 1–20 (2009)
    • 15. Akhatkulov, S., Noorani, M.S.M., Akhadkulov, H.: An analogue of the prime number, Mertens’ and Meissel’s theorems for closed orbits of...
    • 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...
    • 17. Alsharari, F., Noorani, M.S.M., Akhadkulov, H.: Analogues of the prime number theorem and Mertens’ theorem for closed orbits of theMotzkin...
    • 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
    • 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
    • 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...
    • 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...
    • 22. Ward, T.: Ergodic theory: interactions with combinatorics and number theory. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems...
    • 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...
    • 25. Manada, A., Kashyap, N.: On the zeta function of a periodic-finite-type shift. IEICE Trans. Fundamentals Electron. Commun. Comput. Sci....
    • 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...
    • 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...
    • 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
    • 30. Krieger, W., Matsumoto, K.: Zeta functions and topological entropy of the Markov–Dyck shifts. Münster J. Math. 4, 171–184 (2011)
    • 31. Inoue, K., Krieger, W.: Subshifts from sofic shifts and Dyck shifts, zeta functions and topological entropy. arXiv 1–19 (2010). arXiv:1001.1839
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno