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

TL;DR

This paper investigates the conditions under which shifts on multi-dimensional lattices and trees exhibit independence and positive entropy, revealing key differences based on the structure of the underlying trees.

Contribution

It establishes equivalences between positive entropy and independence sets for $ abla^d$ shifts and certain tree-shifts, introducing the boundary independence property for expandable trees.

Findings

Positive entropy iff independence set with positive upper density for $ abla^d$ shifts.

For hereditary shifts, positive entropy corresponds to independence sets on unexpandable trees.

Boundary independence property is equivalent to positive entropy on expandable 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.