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Dendrites of de Groot-Wille type

An example of a rigid dendrite has been roughly described in [J. de Groot et al. 1958]. Its construction is recalled below (see also [Charatonik 1979, Chapter III, Section 5 , p. 227]).

For any i \in \mathbb{N}$ let T_i$ denote the i$-od being the union of i$ straight line unit segments emanating from a point called the origin of T_i$. We proceed by induction. Define Y_0$ as the unit segment. Let x$ be the midpoint of Y_0$, and define Y_1$ as the union of Y_0$ and a diminished copy of T_{j_1} = T_1$ so that the diameter of this copy is less than 1/2$ and that Y_0 \cap
T_1 = \{x\}$. Assume that a tree Y_n$ is defined such that it is the union of finitely many straight line segments and contains (properly diminished) copies of T_1, T_2, \dots,
T_{j_n}$, i.e., of the first j_n$ terms of the sequence \{T_i\}$. To define Y_{n+1}$ consider all maximal free segments in Y_n$. Note that there are finitely many, say m(n)$ of them. Let x$ denote the midpoint of any of these segments. With each point x$ so defined we associate, in a one-to-one way, a set T_i$ taken from the m(n)$ consecutive terms of the sequence \{T_i\}$, i.e., we use in this step of the construction the sets T_{j_n + 1}, T_{j_n
+ 2}, \dots, T_{j_{n + 1}}$, where j_{n+1} = j_n + m(n)$. We take each midpoint as the origin of a properly diminished copy of T_i$, where i \in \{j_n + 1, j_n + 2, \dots, j_{n
+ 1}\}$ in such a way that the diameter of the copy of T_i$ is less than 1/2^{n+1}$ and that Y_n$ has only the point x$ in common with the attached copy of T_i$. All this can clearly be done so carefully that the resulting set Y_{n+1}$ is a tree and the limit continuum

\displaystyle Y = \mathrm{cl}\,(\bigcup \{Y_n: n \in \mathbb{N}\})$

is a dendrite. See Figure A.

Figure 1.3.15: ( A ) Y_3$

Figure: ( AA ) Y_3$

The following properties of the dendrite Y$ are shown in [Charatonik 1979, Statement 10, p. 229].

  1. The set of all end points of Y$ is dense in Y$.
  2. For each n \in \mathbb{N}$ the set of all points of Y$ that are of order at least n$ is dense in Y$.
  3. For each natural n \ge 3$ the dendrite Y$ contains exactly one point of order n$.
  4. For each point p \in Y$ and for each component C$ of Y
\setminus \{p\}$ the continuum C \cup \{p\}$ contains a homeomorphic copy of Y$.
  5. Y$ is not strongly rigid.
  6. Y$ is chaotic.

Recall that for any dendrite conditions 1 and 3 imply 5 and 6, see [Charatonik 1999, Theorem 13, p. 22]. However, neither the above mentioned conditions, nor a rough description given in [J. de Groot et al. 1958], nor the one presented in [Charatonik 1979] lead to a uniquely determined dendrite, because the constructed dendrite Y$ depends on a function that assigns the consecutive i$-ods T_i$ (used in the succesive steps of the construction) to the midpoints of the maximal free arcs in the trees Y_n$. Thus we refer to any of the dendrites Y$ obtained in this way as to a dendrite of de Groot-Wille type .

An example of a dendrite of de Groot-Wille type that is chaotic but not openly chaotic is constructed in [Charatonik 2000, Theorem 3.10, p. 646].

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: Modified Miller dendrites Up: Dendrites Previous: Miller dendrite
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih