Abstract
In this paper, we introduce a new type of F-equicontinuity, which is the opposite of the existence of kF-sensitive pairs almost everywhere and show an analogue of Auslander-Yorke dichotomy theorem. Precisely, under the condition that kF is a translation invariant family, we prove that a transitive system either has F-sensitive pairs almost everywhere or is almost kF-equicontinuous of new type. Also we show that the new type of F-equicontinuity is preserved by an open factor map and consider the implication between the new type of F-equicontinuity and mean equicontinuity.
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