The depth of centres of maps on dendrites

Abstract

AbstractXiong proved that if f: I → I is any map of the unit interval I, then the depth of the centre of f is at most 2, and Ye proved that for any map f: T → T of a finite tree T, the depth of the centre of f is at most 3. It is natural to ask whether the result can be dendrites. In this note, we show that there is dendrite D such that for any countable ordinal number λ there is a map f: D →D such that the depth of centre of f is λ. As a corollary, we show that for any countable ordinal number λ there is a map (respectively a homeomorphism) f of a 2-dimensional ball B2 (respectively a 3-dimensional ball B3) such that the depth of centre of f is λ.