### 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 language | English |
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Title of host publication | Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018 |

Editors | Mohd Hafiz Mohd, Norazrizal Aswad Abdul Rahman, Nur Nadiah Abd Hamid, Yazariah Mohd Yatim |

Publisher | Springer |

Pages | 147-171 |

Number of pages | 25 |

ISBN (Print) | 9789813298316 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

Event | SEAMS School on Dynamical Systems and Bifurcation Analysis, DySBA 2018 - George Town, Malaysia Duration: 6 Aug 2018 → 13 Aug 2018 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 295 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | SEAMS School on Dynamical Systems and Bifurcation Analysis, DySBA 2018 |
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Country | Malaysia |

City | George Town |

Period | 6/8/18 → 13/8/18 |

### Fingerprint

### 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

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Counting Closed Orbits in Discrete Dynamical Systems

AU - Nordin, Azmeer

AU - Noorani, Mohd Salmi Md

AU - Dzul-Kifli, Syahida Che

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Artin-Mazur zeta function

KW - Countable state Markov shift

KW - Dyck shift

KW - Meissel’s orbit theorem

KW - Mertens’ orbit theorem

KW - Motzkin shift

KW - Prime orbit theorem

KW - Shift of finite type

UR - http://www.scopus.com/inward/record.url?scp=85075564377&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075564377&partnerID=8YFLogxK

U2 - 10.1007/978-981-32-9832-3_9

DO - 10.1007/978-981-32-9832-3_9

M3 - Conference contribution

AN - SCOPUS:85075564377

SN - 9789813298316

T3 - Springer Proceedings in Mathematics and Statistics

SP - 147

EP - 171

BT - Dynamical Systems, Bifurcation Analysis and Applications, DySBA 2018

A2 - Mohd, Mohd Hafiz

A2 - Abdul Rahman, Norazrizal Aswad

A2 - Abd Hamid, Nur Nadiah

A2 - Mohd Yatim, Yazariah

PB - Springer

ER -