Abstract
For a discrete dynamical system, the following functions: (i) prime orbit counting function, (ii) Mertens’ orbit counting function, and (iii) Meissel’s orbit sum, describe the different aspects of the growth in the number of closed orbits of the system. These are analogous to counting functions for primes in number theory. The asymptotic behaviour of those functions can be determined by two approaches: by (i) Artin-Mazur zeta function, or (ii) number of periodic points per period. In the first approach, the analyticity and non-vanishing property of the zeta function lead to the asymptotic equivalence of the prime orbit and Mertens’ orbit counting functions. In the second approach, the estimate on the number of periodic points per period is used to obtain the order of magnitude of all those counting functions. This chapter will introduce the counting functions and demonstrate both approaches in some categories of shift spaces, such as shifts of finite type, countable state Markov shifts, Dyck shifts and Motzkin shifts.
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Acknowledgements
The authors would like to acknowledge the grants FRGS/1/2017/STG06/UKM/01/1 and FRGS/1/2017/STG06/UKM/02/2 by Ministry of Higher Education, Malaysia, and DIP-2017-011 by Universiti Kebangsaan Malaysia for financial support in this research.
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Nordin, A., Noorani, M.S.M., Dzul-Kifli, S.C. (2019). 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. DySBA 2018. Springer Proceedings in Mathematics & Statistics, vol 295. Springer, Singapore. https://doi.org/10.1007/978-981-32-9832-3_9
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