On the independence of shifts defined on ${\mathbb{N}}^d$ and trees

Abstract

In this paper, we study the independence of shifts defined on ${\mathbb{N}}^d$ (${\mathbb{N}}^d$ shift) and trees (tree-shift). Firstly, for the completeness of the article, we provide a proof that an ${\mathbb{N}}^d$ shift has positive (topological) entropy if and only if it has an independence set with positive upper density. Secondly, we obtain that when the base shift $X$ is a hereditary shift, then the associated tree-shift ${\mathcal{T}}_X$ on an unexpandable tree has positive entropy if and only if it has an independence set with positive density. However, the independence of the tree-shift on an expandable tree differs from that of ${\mathbb{N}}^d$ shifts or tree-shifts on unexpandable trees. The boundary independence property is introduced and we prove that it is equivalent to the positive entropy of a tree-shift on an expandable tree.