Directional weak mixing and sequence entropy n-tuples for a measure for $\mathbb{Z}^q$-actions

In this paper, directional weak mixing systems are defined and the directional version of Koopman-von Neumann spectrum mixing theorem is deduced. The relation between directional weak mixing systems and classical weak mixing systems is given. Moreover, the notion of directional sequence entropy n-tuples for a measure is introduced and their properties are investigated. Meanwhile, we describle directional discrete spectrum systems and weak mixing systems via directional sequence entropy $2$-tuple. Finally, directional M-null systems and M-supe systems are defined and studied.