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## Omiljanowski dendrite

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 and call it a dendrite . See Figure A. The Omiljanowski dendrite has the following properties, [Charatonik 1991, Example 6.9, p. 182].

1. All ramification points of are of order 3.
2. The set of all ramification points of is discrete (thus nowhere dense).
3. The set of all end points of is nowhere dense.
4. For each maximal arc in the closure of the set contains a homeomorphic copy of the Cantor set.
5. is monotonely equivalent to the dendrite .
6. The following conditions are equivalent for a dendrite (see [Charatonik 1991, Theorem 6.14, p. 185] and [Charatonik et al. 1994, Theorem 5.35, p. 17]; compare [Charatonik et al. 1998, Theorem 2.20, p. 238]):
1. is monotonely equivalent to ;
2. is monotonely equivalent to ;
3. is monotonely equivalent to for each ;
4. is monotonely equivalent to for each nonempty set ;
5. is monotonely equivalent to every dendrite having dense set of its ramification points;
6. is monotonely equivalent to some dendrite having dense set of its ramification points;
7. contains a homeomorphic copy of every dendrite such that its set of ramification points is discrete and consists of points of order exclusively;
8. contains a homeomorphic copy of the dendrite .

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]).

7. If for a dendrite the set of its ramification points is dense in , then is monotonely homogeneous.

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].

8. If a dendrite contains a homeomorphic copy of , then is monotonely homogeneous.

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.     Next: Chaotic and rigid dendrites Up: Dendrites Previous: Monotone equivalence and monotone
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30