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

Abstract

In this paper, we introduce an $\mathcal{F}$-equi\-con\-ti\-nui\-ty and show an analogue of Auslander-Yorke dichotomy theorem for $\mathcal{F}$-sensitivity. Precisely, under the condition that $k\mathcal{F}$ is translation invariant, we prove that a transitive system is either $\mathcal{F}$-sensitive or almost $k\mathcal{F}$-equi\-con\-ti\-nuo\-us , and so generalize the result of previous work. Also we show that $\mathcal{F}$-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.