Transitivity And Related Notions For Graph Induced Symbolic Systems

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 $(HV{)}_{ij}\ne 0\iff (VH{)}_{ij}\ne 0$ (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 $H^r{V}^s$. 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.