标题： 关于定义在$\mathbb{N}^d$上的平移的独立性以及树 显示英文标题

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

摘要： 在本文中，我们研究定义在$\mathbb{N}^d$（$\mathbb{N}^d$移位）和树（树移位）上的移位的独立性。 首先，为了文章的完整性，我们提供一个证明，即一个$\mathbb{N}^d$移位具有正（拓扑）熵当且仅当它有一个具有正上密度的独立集。 其次，我们得到当基移位$X$是一个遗传移位时，关联在不可扩展树上的树移位$\mathcal{T}_X$具有正熵当且仅当它有一个具有正密度的独立集。 然而，在可扩展树上的树移位的独立性与$\mathbb{N}^d$移位或在不可扩展树上的树移位的独立性不同。 引入了边界独立性性质，并证明它与可扩展树上的树移位的正熵等价。 显示英文摘要

摘要： 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.