Vol. 64 Part 1 (1998)
The depth of centres of maps on dendrites
Hisao Kato
Institute of Mathematics
University of Tsukuba
Ibaraki 305
Japan
e-mail: hisakato@sakura.cc.tsukuba.ac.jp
Abstract:
Xiong proved that if 
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 
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 
there is a map 
such that the depth of
centre of f is .
As a corollary, we show that for any countable ordinal number
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
.
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
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
there is a map
such that the depth of
centre of f is .
As a corollary, we show that for any countable ordinal number
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
.
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© Copyright 1998, Australian Mathematical Society Greg Lewis
1998-02-18