Directional weak mixing and applications

We introduce and characterize the concept of directional weak mixing through independence, sequence entropy, and other notions. In particular, we obtain a directional version of the mean ergodic theorem. As an application, we establish a law of large numbers. Additionally, we deduce a directional version of the Koopman–von Neumann spectrum mixing theorem. Furthermore, we explore the relation between directional weak mixing and weak mixing.