The Depth of Centres of Maps on Dendrites

Xiong 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 generalized to maps of dendrites. In this note, we show that there is a dendrite D such that for any countable ordinal number 1⁄2 there is a map f : D! D such that the depth of centre of f is 1⁄2. As a corollary, we show that for any countable ordinal number 1⁄2 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 1⁄2. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 54H20; secondary 58F50.