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].
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.