THE DEPTH OF CENTERS OF MAPS ON DENDRITES

Xiong proved that if S : I --> I is any map of the unit interval I, th en the depth of the centre of S is at most 2, and Ye proved that for a ny 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 lambda there isa map f : D --> D such that the depth of centre of f is h. As a corollary, we sh ow that for any countable ordinal number lambda there is a map (respec tively a homeomorphism) f of a 2-dimensional ball B-2 (respectively a 3-dimensional ball B-3) such that the depth of centre of f is lambda.