Computations on Sofic S-gap Shifts

TL;DR

This paper investigates the algebraic and dynamical properties of sofic S-gap shifts, providing explicit formulas for their characteristic polynomials, zeta functions, and Bowen-Franks groups, thereby advancing the understanding of their structure.

Contribution

It explicitly determines the characteristic polynomial and zeta function formulas for sofic S-gap shifts, and computes their Bowen-Franks groups, offering new algebraic insights.

Findings

Explicit formulas for characteristic polynomials of adjacency matrices.

Closed-form expressions for zeta functions of S-gap shifts.

Calculation of Bowen-Franks groups for sofic S-gap shifts.

Abstract

Let $S=\{s_{n}\}$ be an increasing finite or infinite subset of $\mathbb N \bigcup \{0\}$ and $X(S)$ the $S$-gap shift associated to $S$. Let $f_{S}(x)=1-\sum\frac{1}{x^{s_{n}+1}}$ be the entropy function which will be vanished at $2^{h(X(S))}$ where $h(X(S))$ is the entropy of the system. Suppose $X(S)$ is sofic with adjacency matrix $A$ and the characteristic polynomial $\chi_{A}$. Then for some rational function $ Q_{S} $, $\chi_{A}(x)=Q_{S}(x)f_{S}(x)$. This $ Q_{S} $ will be explicitly determined. We will show that $\zeta(t)=\frac{1}{f_{S}(t^{-1})}$ or $\zeta(t)=\frac{1}{(1-t)f_{S}(t^{-1})}$ when $|S|<\infty$ or $|S|=\infty$ respectively. Here $\zeta$ is the zeta function of $X(S)$. We will also compute the Bowen-Franks groups of a sofic $S$-gap shift.