The
Omiljanowski dendrite
has been constructed by K. Omiljanowski in
[Charatonik 1991, Example 6.9, p. 182] (see also [Charatonik 1991, Remark 6.11 and Theorem 6.12,
p. 183]). It is defined as the closure of the union of an increasing
sequence of dendrites in the plane. We start with the unit straight line
segment denoted by . Divide
into three equal subsegments and in
the middle of them,
, locate a thrice diminished copy of the Cantor
ternary set
. At the midpoint of each contiguous interval
to
(i.e., a component
of
) we erect perpendicularly to
a straight line segment whose length equals length of
. Denote by
the union of
and of all erected segments (there are countably many of
them, and their lengths tend to zero). We perform the same construction on
each of the added segments: divide such a segment into three equal parts,
locate in the middle part
a copy of the Cantor set
properly
diminished, at the midpoint of any component
of
construct a perpendicular to
segment as long as
is, and denote by
the union of
and of all attached segments. Continuing in this
manner we get a sequence of dendrites
. Finally we put
The Omiljanowski dendrite has the following properties, [Charatonik 1991, Example
6.9, p. 182].
According to Property 7
in 1.3.8 (see also
Property 6 in 1.3.7)
any dendrite is monotonely homogeneous. Another, less restrictive, sufficient
condition for monotone homogeneity of a dendrite is the
following (see [Charatonik et al. 1997a, Proposition 15, p.
364]).
The condition
, being sufficient, is far from
being necessary. Namely the Omiljanowski dendrite
has
the set
discrete, and it is monotonely homogeneous. Moreover, the following
statement holds, [Charatonik et al. 1997a, Proposition 20, p.
366].
It would be interesting to know if the converse to the above statement holds
true, i.e., if containing the dendrite characterizes monotonely
homogeneous dendrites. In other words, we have the following question.
Question. Does every monotonely homogeneous
dendrite contain a homeomorphic copy of ?
The above question is closely related to a more general problem.
Problem. Give any structural characterization of monotonely homogeneous.
Here you can find source files of this example. Here you can check the table of properties of individual continua. Here you can read Notes or write to Notes ies of individual continua.