For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

中文翻译：

---

主代数作用的玻尔混沌性和 Riesz 积测度

对于连续的 $\mathbb {N}^d$ 或者 $\mathbb {Z}^d$ 为了解决紧空间上的作用，我们引入了玻尔混沌性的概念，它是拓扑共轭的不变量，并且被证明比正熵更强。我们证明所有主代数 $\mathbb {Z}$ 正熵的作用是玻尔混沌。对于主代数动作也证明了同样的情况 $\mathbb {Z}^d$ 在可和同宿点存在的条件下具有正熵。