Transitivity And Related Notions For Graph Induced Symbolic Systems

TL;DR

This paper explores the dynamical properties of two-dimensional symbolic systems generated by graphs, focusing on transitivity, mixing, and decomposability, with criteria based on adjacency matrices and graph products.

Contribution

It introduces new criteria for transitivity and mixing in 2D shift spaces using adjacency matrices, and analyzes graph decomposability into simpler components.

Findings

Doubly transitive shifts imply the same for the generated 2D shift.

Directional transitivity characterized via block representations of matrix powers.

Necessary and sufficient conditions for horizontal and vertical transitivity.

Abstract

In this paper, we investigate the dynamical behavior of a two dimensional shift $X_G$ (generated by a two dimensional graph $G=(\mathcal{H},\mathcal{V})$) using the adjacency matrices of the generating graph $G$. In particular, we investigate properties such as transitivity, directional transitivity, weak mixing, directional weak mixing and mixing for the shift space $X_G$. We prove that if $(HV)_{ij}\neq 0 \Leftrightarrow (VH)_{ij}\neq 0$ (for all $i,j$), while doubly transitivity (weak mixing) of $X_H$ (or $X_V$) ensures the same for two dimensional shift generated by the graph $G$, directional transitivity (in the direction $(r,s)$) can be characterized through the block representation of $H^rV^s$. We provide necessary and sufficient criteria to establish horizontal (vertical) transitivity for the shift space $X_G$. We also provide examples to establish the necessity of the conditions imposed. Finally, we investigate the decomposability of a given graph into product of graphs with reduced complexity.