Directional Kronecker algebra for $\mathbb{Z}^q$-actions

TL;DR

This paper introduces directional sequence entropy and Kronecker algebra for $ obreak bZ^q$-systems, establishing their relations and characterizations of systems with directional discrete spectrum and null properties.

Contribution

It defines and explores directional entropy and Kronecker algebra in $bZ^q$-systems, linking directional discrete spectrum to classical discrete spectrum.

Findings

Directional sequence entropy and Kronecker algebra are introduced.

A $bZ^q$-system has directional discrete spectrum iff it is directional null.

Having directional discrete spectrum along $q$ independent directions implies classical discrete spectrum.

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, direcitonal 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.