Directional Pinsker algebra and its applications

TL;DR

This paper introduces the directional Pinsker algebra, constructs a skew product to analyze it, and demonstrates its implications for chaos and entropy in $ obreak bZ^2$-systems, revealing new insights into directional measure-theoretic properties.

Contribution

It defines the directional Pinsker algebra, constructs a skew product for its study, and applies it to establish new results on chaos and entropy in $ obreak bZ^2$-systems.

Findings

Positive directional measure-theoretic entropy implies multivariant directional mean Li-Yorke chaos.

The intersection of entropy tuples and asymptotic tuples is dense among entropy tuples.

New framework for analyzing directional properties in $bZ^2$-systems.

Abstract

In this paper, we introduce the directional Pinsker algebra, and construct a skew product to study it. As applications, we show that 1. if a $\mathbb{Z}^2$-system with positive directional measure-theoretic entropy then it is multivariant directional mean Li-Yorke chaotic along the corresponding direction; 2. for any ergodic measure on a $\mathbb{Z}^2$-system, the intersection of the set of directional measure-theoretic entropy tuples with the set of directional asymptotic tuples is dense in the set of directional measure-theoretic entropy tuples.