An analogue of the prime number, Mertens’ and Meissel’s theorems for closed orbits of the Dyck shift

Akhatkulov, Sokhobiddin; Noorani, Mohd Salmi Md; Akhadkulov, Habibulla

doi:10.1063/1.4980971

Cited by 6 publications

(4 citation statements)

References 9 publications

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“…These results are more precise than the previous results obtained via the estimation of number of periodic points in [9,10]. Akhatkulov et al [11] later found sharper results for Dyck shifts, but it did not include similar results for Motzkin shifts.…”

Section: Introductioncontrasting

confidence: 54%

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

Nordin

Noorani

2021

Mathematics

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For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

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Section: Introductioncontrasting

confidence: 54%

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

Nordin

Noorani

2021

Mathematics

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“…In fact, if S = 0 in the above, we obtain the orbit growth for Dyck shift over R pairs of matching symbols (see [9]). Later, Akhatkulov et al [10] obtained sharper results for the Dyck shift, which are…”

Section: Introductionmentioning

confidence: 95%

Orbit Growth of Periodic-Finite-Type Shifts via Artin–Mazur Zeta Function

Nordin

Noorani

2020

Mathematics

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“…While we focus on the approach via zeta function, there are other methods to obtain the orbit growth of a system, such as using orbit Dirichlet series [13], orbit monoids [14] and estimates on the number of periodic points [15][16][17]. Furthermore, similar research problem has been studied for group actions on dynamical systems, and some recent results include the orbit growths of nilpotent group shifts [18], algebraic flip systems [19] and flip systems for shifts of finite type [20].…”

Section: Introductionmentioning

confidence: 99%

A short note on the orbit growth of sofic shifts

Nordin¹,

Noorani²

2022

Preprint

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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 space. 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 analiticity of the Artin-Mazur zeta function of the space. Despite having a closed-form expression, the zeta function is expressed implicitly in terms of several signed subset matrices, and this makes the study on its analyticity to be seemingly difficult. In this paper, we will prove the asymptotic behaviours of the counting functions for a sofic shift via its zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof is rather short and only uses well-known facts about a sofic shift, especially on its minimal right-resolving presentation.

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An analogue of the prime number, Mertens’ and Meissel’s theorems for closed orbits of the Dyck shift

Cited by 6 publications

References 9 publications

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

Orbit Growth of Periodic-Finite-Type Shifts via Artin–Mazur Zeta Function

A short note on the orbit growth of sofic shifts

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