F-Mean Equicontinuity on Amenable Semigroups

In this paper, we introduce and study concepts of mean equicontinuity and mean sensitivity via Furstenberg family with respect to a countable left amenable semigroup S , We also present an analogue of the Auslander–Yorke dichotomy, demonstrating that a transitive system either has F -mean sensitive pair almost everywhere or is almost -mean equicontinuous. Also, we prove that the definition of F -mean equicontinuity is preserved by an open factor map.