Auslander-Yorke dichotomy theorem for indecomposable systems

Abstract

The Auslander-Yorke dichotomy theorem says that every transitive system is either sensitive or almost equicontinuous. A topological dynamical system $(X,f)$ is said to be indecomposable if every two $f$- invariant closed subsets having non-empty interiors must have common interior points. This notion is strictly weaker than transitivity. In this paper, we show that every indecomposable system with $f$ being semi-open is either almost equicontinuous or sensitive, which complements the Auslander-Yorke dichotomy theorem. Moreover, we prove that in an almost equicontinuous indecomposable system all equicontinuous points have the same $\omega$- limited set.