Counting Closed Orbits in Discrete Dynamical Systems

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

doi:10.1007/978-981-32-9832-3_9

Cited by 5 publications

(5 citation statements)

References 18 publications

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“…The results include the cases for Dyck shifts, Motzkin shifts and a certain class of shift subspaces from Dyck shifts. Although the approach via zeta function is straightforward due to Theorem 1, the difficulty arises due to the form of its zeta function in (19). Since the zeta function contains square roots of polynomials, some tools in complex analysis are needed to determine the meromorphic extension in (6).…”

Section: Discussionmentioning

confidence: 99%

“…However, our machinery here is the approach via zeta function. For a detailed exposure on the topic of orbit growth, we encourage the interested readers to refer to our survey paper in [19]. We shall also mention that there is a similar research problem of counting finite orbits for group actions, and some asymptotic results have been obtained for finitely-generated torsion-free nilpotent group shifts [20], algebraic flip systems [21] and flip systems of shifts of finite type [22].…”

Section: Introductionmentioning

confidence: 99%

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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: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Nordin

Noorani

2021

Mathematics

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“…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]. Since our introduction on this subject is rather short, we encourage readers to explore those papers above, and additionally the expository chapters by us [21] and Ward [22].…”

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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“…The results shown above are enough to demonstrate the progress in this research interest in recent years. As a supplementary, interested readers may refer to our survey in [14] and the references therein for more exposure on the topic of orbit counting in discrete dynamical systems.…”

Section: Introductionmentioning

confidence: 99%