F-equicontinuity and an Analogue of Auslander-Yorke Dichotomy Theorem

In this paper, we introduce an $\mathcal{F}$-equi-con-ti-nui-ty and show an analogue of Auslander-Yorke dichotomy theorem for ${\mathscr F}$-sensitivity\mathrm{.}Precisely,undertheconditionthat$k{\mathscr F}$ is translation invariant, we prove that a transitive system is either ${\mathscr F}$-sensitiveoralmost$k{\mathscr F}$-equi-con-ti-nuo-us , and so generalize the result of previous work. Also we show that ${\mathscr F}$-equi-con-ti-nui-ty is preserved by an open factor map and consider the implication between $\mathcal{F}$-equi-con-ti-nui-ty and mean equi-con-ti-nui-ty.