A note on the map with the whole space being a scrambled set Original Research Article

In this paper we prove that there is no compact system with the whole space being a Schweizer–Smital scrambled set and the Huang–Ye homeomorphism [W. Huang, X.D. Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergod. Theory & Dynam. Systems 21 (2001) 77–91] has no Schweizer–Smital pair. Moreover, we present a simple noncompact system whose Schweizer–Smital scrambled set can be the whole space.