Directional Pinsker algebra and its applications

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.