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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 let denote the -od being the union of straight line unit segments emanating from a point called the origin of . We proceed by induction. Define as the unit segment. Let be the midpoint of , and define as the union of and a diminished copy of so that the diameter of this copy is less than and that . Assume that a tree is defined such that it is the union of finitely many straight line segments and contains (properly diminished) copies of , i.e., of the first terms of the sequence . To define consider all maximal free segments in . Note that there are finitely many, say of them. Let denote the midpoint of any of these segments. With each point so defined we associate, in a one-to-one way, a set taken from the consecutive terms of the sequence , i.e., we use in this step of the construction the sets , where . We take each midpoint as the origin of a properly diminished copy of , where in such a way that the diameter of the copy of is less than and that has only the point in common with the attached copy of . All this can clearly be done so carefully that the resulting set is a tree and the limit continuum

is a dendrite. See Figure A.

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

1. The set of all end points of is dense in .
2. For each the set of all points of that are of order at least is dense in .
3. For each natural the dendrite contains exactly one point of order .
4. For each point and for each component of the continuum contains a homeomorphic copy of .
5. is not strongly rigid.
6. 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 depends on a function that assigns the consecutive -ods (used in the succesive steps of the construction) to the midpoints of the maximal free arcs in the trees . Thus we refer to any of the dendrites 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: Modified Miller dendrites Up: Dendrites Previous: Miller dendrite
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30