On Graph Induced Symbolic Systems

TL;DR

This paper explores the structure of multi-dimensional shift spaces generated by graphs, establishing conditions for periodic points, finiteness, and the use of permutation matrices, advancing symbolic dynamics theory.

Contribution

It demonstrates that all finite type shifts can be generated by graphs and characterizes periodicity and finiteness using generating matrices.

Findings

A $k$-dimensional shift of finite type can be generated by a $k$-dimensional graph.

A 2D shift space has a periodic point if and only if it has an $(m,n)$-periodic point.

A shift space is finite iff it is generated by permutation matrices.

Abstract

\begin{abstract} In this paper, we investigate a shift arising from graph $G$. We prove that any $k$-dimensional shift of finite type can be generated through a $k$-dimensional graph. We investigate the structure of the shift space using the generating matrices for the shift space. We prove that a two dimensional shift space has a horizontally (vertically) periodic point if and only if it possesses a $(m,n)$-periodic point (for some $m,n\in \mathbb{Z}\setminus \{0\}$). We prove that a shift space is finite if and only if it can be generated by permutation matrices. We study the non-emptiness problem and existence of periodic points in terms of the generating matrices.