Monotone homogeneity of dendrites
by
Janusz J. Charatonik
University of Wroclaw, Universidad Nacional Autonoma de Mexico
Coauthors: Wlodzimierz J. Charatonik

A dendroid means an arcwise connected and hereditarily unicoherent continuum. A dendroid which is locally connected is a dendrite. Let M be a class of mappings such that all homeomorphisms are in M and the composition of any two mappings in M is also in M. A continuum X is said to be M-homogeneous provided that for every two points p and q of X there exists a surjective mapping f: X --> X such that f(p) = q and f is in M. We take as M the class of monotone mappings (i.e., with connected point inverses) between continua. This class is contained in the class of confluent mappings (i.e. such f: X --> Y that if Q is a subcontinuum of the range Y and C is a component of f^-1(Q), then f(C)=Q).

QUESTION 1. What dendrites (dendroids) are monotone homogeneous?

It is known that a dendrite is monotone homogeneous if and only if it is confluent homogeneous. Is this equivalence valid for dendroids?

H. Kato, [On problems of H. Cook, Topology Appl. 26 (1987), 219-228] proved that the standard universal dendrite D₃ of order 3 (i.e. such that all its ramification points are of order 3 and the set of these points is dense in the dendrite) is monotone homogeneous.

THEOREM 1. If there exists a monotone mapping from a dendrite X onto D₃, then X is monotone homogeneous.

COROLLARY. If a dendrite has a dense set of its ramification points, then it is monotone homogeneous.

The converse is not true since there is a monotone homogeneous dendrite L₀ such that all its ramification points are of order 3 and the set of these points is nowhere dense (J. J. Charatonik, Monotone mappings ofuniversal dendrites, Topology Appl. 38 (1991), 163-187).

THEOREM 2. If a dendrite contains the dendrite L₀, then it is monotone homogeneous.

QUESTION 2. Is the converse to the above implication true?

A dendroid X is said to be smooth if there is a point p in X such that for every point x in X and for every sequence { x_n } of points converging to x the sequence of arcs px_n converges to the arc px.

THEOREM 3. If a smooth dendroid is monotone homogeneous, then it is a dendrite.

QUESTION 3. Is smoothness an essential assumption in the above result?

Date received: February 14, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caab-78.