Counting Closed Orbits in Discrete Dynamical Systems

Azmeer Nordin, Mohd Salmi Md Noorani, Syahida Che Dzul-Kifli

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish
Title of host publicationDynamical Systems, Bifurcation Analysis and Applications, DySBA 2018
EditorsMohd Hafiz Mohd, Norazrizal Aswad Abdul Rahman, Nur Nadiah Abd Hamid, Yazariah Mohd Yatim
PublisherSpringer
Pages147-171
Number of pages25
ISBN (Print)9789813298316
DOIs
Publication statusPublished - 1 Jan 2019
EventSEAMS School on Dynamical Systems and Bifurcation Analysis, DySBA 2018 - George Town, Malaysia
Duration: 6 Aug 201813 Aug 2018

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume295
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceSEAMS School on Dynamical Systems and Bifurcation Analysis, DySBA 2018
CountryMalaysia
CityGeorge Town
Period6/8/1813/8/18

Fingerprint

Closed Orbit
Counting Function
Discrete Dynamical Systems
Counting
Orbit
Periodic Points
Riemann zeta function
Shift of Finite Type
Asymptotic Equivalence
Analyticity
Number theory
Countable
Asymptotic Behavior
Estimate
Demonstrate

Keywords

  • Artin-Mazur zeta function
  • Countable state Markov shift
  • Dyck shift
  • Meissel’s orbit theorem
  • Mertens’ orbit theorem
  • Motzkin shift
  • Prime orbit theorem
  • Shift of finite type

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Nordin, A., Noorani, M. S. M., & Dzul-Kifli, S. C. (2019). Counting Closed Orbits in Discrete Dynamical Systems. In M. H. Mohd, N. A. Abdul Rahman, N. N. Abd Hamid, & Y. Mohd Yatim (Eds.), Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018 (pp. 147-171). (Springer Proceedings in Mathematics and Statistics; Vol. 295). Springer. https://doi.org/10.1007/978-981-32-9832-3_9

Counting Closed Orbits in Discrete Dynamical Systems. / Nordin, Azmeer; Noorani, Mohd Salmi Md; Dzul-Kifli, Syahida Che.

Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018. ed. / Mohd Hafiz Mohd; Norazrizal Aswad Abdul Rahman; Nur Nadiah Abd Hamid; Yazariah Mohd Yatim. Springer, 2019. p. 147-171 (Springer Proceedings in Mathematics and Statistics; Vol. 295).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Nordin, A, Noorani, MSM & Dzul-Kifli, SC 2019, Counting Closed Orbits in Discrete Dynamical Systems. in MH Mohd, NA Abdul Rahman, NN Abd Hamid & Y Mohd Yatim (eds), Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018. Springer Proceedings in Mathematics and Statistics, vol. 295, Springer, pp. 147-171, SEAMS School on Dynamical Systems and Bifurcation Analysis, DySBA 2018, George Town, Malaysia, 6/8/18. https://doi.org/10.1007/978-981-32-9832-3_9
Nordin A, Noorani MSM, Dzul-Kifli SC. Counting Closed Orbits in Discrete Dynamical Systems. In Mohd MH, Abdul Rahman NA, Abd Hamid NN, Mohd Yatim Y, editors, Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018. Springer. 2019. p. 147-171. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-981-32-9832-3_9
Nordin, Azmeer ; Noorani, Mohd Salmi Md ; Dzul-Kifli, Syahida Che. / Counting Closed Orbits in Discrete Dynamical Systems. Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018. editor / Mohd Hafiz Mohd ; Norazrizal Aswad Abdul Rahman ; Nur Nadiah Abd Hamid ; Yazariah Mohd Yatim. Springer, 2019. pp. 147-171 (Springer Proceedings in Mathematics and Statistics).
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