$\mathscr{F}$-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 $\mathscr{F}$-mean sensitive pair almost everywhere or is almost $\kappa\mathscr{F}$-mean equicontinuous. Also, we prove that the definition of $\mathscr{F}$-mean equicontinuity is preserved by an open factor map.