Directional Kronecker algebra for -actions

Abstract

In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb {Z}^q$ -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb {Z}^q$ -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb {Z}^q$ -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

Directional stable sets and mean Li-Yorke chaos in positive directional entropy systems

“…x, {nβ + t}), where C is the Borel σ-algebra on [0, 1) and m is the Lebesgue measure on [0, 1). Recently, two authors in this paper [21] showed that the product σ-algebra of the directional Kronecker algebra of X and the Borel σ-algebra of [0, 1) is equal to the Kronecker algebra of X. We remark that the result above does not omit the second coordinate in the original proof but from the proof, it is easy to see the result also holds when we omit the second coordinate.…”

“…in this paper. Recently, two authors in this paper [21] showed the product σ-algebra of the directional Kronecker algebra of the original Z 2 -m.p.s. and the Borel σ-algebra of [0, 1) is equal to the Kronecker algebra of its suspension system, and used this result to study the relation between directional properties and spectrum.…”

Sequence entropy for amenable group actions*

“…As an application, they characterized compact, rigid and mixing extensions via conditional sequence entropy. More recently, directional sequence entropy was introduced to study the directional discrete spectrum by Liu and Xu [16] and directional weak mixing by Liu [17].…”

Directional ergodicity and weak mixing for actions of $\mathbb R^d$ and $\mathbb Z^d$

“…However, they are different for directional rigidity and the question for recurrence in the case of groups Γ as described above remains open [8]. More recently Liu and Xu ([19], [20]) have defined directional properties intrinsically and studied their relationship to spectrum.…”