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

Nordin, Azmeer; Noorani, Mohd Salmi Md

doi:10.3390/math8050685

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…In the literature, this approach was used to determine the orbit growths of ergodic toral automorphisms [5,6] and several types of shift spaces. These include shifts of finite type [2,7], periodic-finite-type shifts [4], Dyck and Motzkin shifts [8], and bouquet-Dyck shifts [9]. Similar results can be deduced for beta shifts [10], negative beta shifts [11] and shifts of quasi-finite type [12], albeit these findings are not stated in their respective papers.…”

Section: Introductionmentioning

confidence: 54%

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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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Section: Introductionmentioning

confidence: 54%

“…A periodic-finite-type shift (which includes a shift of finite type) is a sofic shift. We obtained its orbit growth in [4] by using its Moision-Siegel presentation [24,25]. However, our previous work was incomplete since the result was only applicable for the one with an irreducible such presentation.…”

Section: Orbit Growth Of a Periodic-finite-type Shiftmentioning

confidence: 99%

A short note on the orbit growth of sofic shifts

Nordin¹,

Noorani²

2022

Preprint

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“…In recent years, the above approach had been demonstrated to obtain the orbit growth of periodic-finite-type shifts [7], and also Dyck and Motzkin shifts [8]. However, the tools used to analyse their respective zeta function are different for each case.…”

Section: Introductionmentioning

confidence: 99%

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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“…This result was obtained through a generating function called Artin-Mazur zeta function [5]. Apart from shifts of finite type, there were results for other systems that utilized this approach via their zeta function, such as ergodic toral automorphisms [21], periodic-finite-type shifts [14], and Dyck and Motzkin shifts [16]. In fact, different approaches were also introduced to obtain such results for other systems, such as using estimates of the number of periodic points on Dyck and Motzkin shifts [2,3,4], counting in orbit monoids [17] and using orbit Dirichlet series on some algebraic systems [6,7].…”

mentioning

confidence: 99%

Counting finite orbits for the flip systems of shifts of finite type

Nordin¹,

Noorani²

2021

DCDS

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For a discrete system (X, T), the flip system (X, T, F) can be regarded as the action of infinite dihedral group D∞ on the space X. Under this action, X is partitioned into a set of orbits. We are interested in counting the finite orbits in this partition via the prime orbit counting function. In this paper, we prove the asymptotic behaviour of this counting function for the flip systems of shifts of finite type. The proof relies mostly on combinatorial calculations instead of the usual approach via zeta function. Here, we are able to obtain more precise asymptotic result for this D∞-action on shifts of finite type as compared to other group actions on systems available in the literature.

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

Cited by 4 publications

References 19 publications

A short note on the orbit growth of sofic shifts

A short note on the orbit growth of sofic shifts

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

Counting finite orbits for the flip systems of shifts of finite type

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