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