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

Azmeer Nordin,Mohd Salmi Md Noorani

doi:10.3390/math9111268

Abstract

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.

Highlights

Let (X, T) be a discrete dynamical system, where X is a topological space and T : X → X is a continuous map
The orbit growths of Dyck and Motzkin shifts can be deduced by setting P = Q = 0 and Q = 1, respectively, and calculating the exact value of φ based on Theorem 4
The results include the cases for Dyck shifts, Motzkin shifts and a certain class of shift subspaces from Dyck shifts

Summary

Introduction

Let (X, T) be a discrete dynamical system, where X is a topological space and T : X → X is a continuous map. Certain types of shift spaces in the literature have been shown to have a non-vanishing meromorphic extension of their respective zeta function, and this implies the orbit growth as in Theorem 1. Those results on orbit growth are not stated therein. The aim of our paper is to obtain the orbit growth of the bouquet-Dyck shifts via their respective zeta function. We find the meromorphic extension of the zeta function, and deduce the orbit growth in Theorem 4 We demonstrate these results on the shift spaces found in [26] as an example

Bouquet-Dyck Shifts

Orbit Growth of Bouquet-Dyck Shifts

Conclusions