Multi-transitivity and Δ-transitivity for semigroup actions

In this paper, we study multi-transitivity and Δ-transitivity for semigroup actions and give several characterizations of them, which generalize related results in the literature. We also study Δ-mixing and show that under some conditions a dynamical system is Δ-transitive if and only if it is Δ-mixing with respect to a thick set.