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

中文翻译：

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F-equicontinuity 和 Auslander-Yorke 二分定理的类比

在本文中，我们引入了 ${\mathscr F}$-equi\-con\-ti\-nui\-ty 并展示了 Auslander-Yorke 二分定理对 ${\mathscr F}$ 敏感性的模拟。准确地说，在 $k{\mathscr F}$ 是平移不变的条件下，我们证明了一个传递系统要么是 ${\mathscr F}$ 敏感的，要么几乎是 $k{\mathscr F}$-equi\-con \-ti\-nuo\-us ，因此概括了以前工作的结果。我们还表明 ${\mathscr F}$-equi\-con\-ti\-nui\-ty 由开放因子映射保留，并考虑 ${\mathscr F}$-equi\-con\ 之间的含义-ti\-nui\-ty 和平均 equi\-con\-ti\-nui\-ty。