  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.
