Analogues of Auslander–Yorke theorems for multi-sensitivity

Abstract

We study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. Our main result states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. This is an analog of the Auslander–Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous. Furthermore, we introduce the concept of a syndetically equicontinuous point, and we prove that a transitive system is either thickly sensitive or contains syndetically equicontinuous points, which is a refinement of another well-known result of Akin, Auslander and Berg.