A short note on the orbit growth of sofic shifts

TL;DR

This paper investigates the growth of closed orbits in sofic shifts by analyzing their zeta functions, revealing asymptotic behaviors through properties of associated matrices, with a concise proof leveraging known shift properties.

Contribution

It provides a new, concise proof of the asymptotic behavior of orbit counting functions in sofic shifts using their zeta functions and matrix properties.

Findings

Asymptotic behaviors of orbit counting functions are established.

The proof relies on properties of minimal right-resolving presentations.

Analyticity of the zeta function is linked to matrix characteristics.

Abstract

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.