Analogues of Auslander-Yorke theorems for multi-sensitivity

In this paper we study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. We show that multi-sensitivity implies thick sensitivity, and the converse holds true for transitive systems. Our main result is an analog of the Auslander-Yorke dichotomy theorem: a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. Furthermore, we refine it by introducing the concept of syndetically equicontinuous points: a transitive system is either thickly sensitive or contains syndetically equicontinuous points.