next up previous contents index
Next: Chaotic and rigid dendrites Up: Dendrites Previous: Monotone equivalence and monotone

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 L_1$. Divide L_1$ into three equal subsegments and in the middle of them, M$, locate a thrice diminished copy of the Cantor ternary set C$. At the midpoint of each contiguous interval K$ to C$ (i.e., a component K$ of M \setminus C$) we erect perpendicularly to L_1$ a straight line segment whose length equals length of K$. Denote by L_2$ the union of L_1$ 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 M$ a copy of the Cantor set C$ properly diminished, at the midpoint of any component K$ of M \setminus C$ construct a perpendicular to K$ segment as long as K$ is, and denote by L_3$ the union of L_2$ and of all attached segments. Continuing in this manner we get a sequence of dendrites L_1 \subset L_2 \subset L_3 \subset \dots $. Finally we put

\displaystyle L_0 = \mathrm{cl}\, (\bigcup \{L_i: i \in \Bbb N\}) $

and call it a dendrite L_0$ . See Figure A.

Figure 1.3.12: ( A ) dendrite L_0$
A.gif

The Omiljanowski dendrite L_0$ has the following properties, [Charatonik 1991, Example 6.9, p. 182].

  1. All ramification points of L_0$ are of order 3.
  2. The set R(L_0)$ of all ramification points of L_0$ is discrete (thus nowhere dense).
  3. The set of all end points of L_0$ is nowhere dense.
  4. For each maximal arc A$ in L_0$ the closure of the set A \cap R(L_0)$ contains a homeomorphic copy of the Cantor set.
  5. L_0$ is monotonely equivalent to the dendrite D_3$.
  6. The following conditions are equivalent for a dendrite X$ (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. X$ is monotonely equivalent to D_3$;
    2. X$ is monotonely equivalent to D_\omega $;
    3. X$ is monotonely equivalent to D_m$ for each m \in \{3,4,\dots,\omega\}$;
    4. X$ is monotonely equivalent to D_S$ for each nonempty set S \subset \{3, 4, \dots, \omega\}$;
    5. X$ is monotonely equivalent to every dendrite Y$ having dense set R(Y)$ of its ramification points;
    6. X$ is monotonely equivalent to some dendrite Y$ having dense set R(Y)$ of its ramification points;
    7. X$ contains a homeomorphic copy of every dendrite L$ such that its set R(L)$ of ramification points is discrete and consists of points of order 3$ exclusively;
    8. X$ contains a homeomorphic copy of the dendrite L_0$.

    According to Property 7 in 1.3.8 (see also Property 6 in 1.3.7) any dendrite D_S$ 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 X$ the set R(X)$ of its ramification points is dense in X$, then X$ is monotonely homogeneous.

    The condition \mathrm{cl}\,R(X) = X$, being sufficient, is far from being necessary. Namely the Omiljanowski dendrite L_0$ has the set R(L_0)$ discrete, and it is monotonely homogeneous. Moreover, the following statement holds, [Charatonik et al. 1997a, Proposition 20, p. 366].

  8. If a dendrite X$ contains a homeomorphic copy of L_0$, then X$ is monotonely homogeneous.

It would be interesting to know if the converse to the above statement holds true, i.e., if containing the dendrite L_0$ characterizes monotonely homogeneous dendrites. In other words, we have the following question.

Question. Does every monotonely homogeneous dendrite contain a homeomorphic copy of L_0$?

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