{Dendrites of de Groot-Wille type}

An example of a rigid dendrite has been roughly described in \cite{Groot+1958a}. Its
construction is recalled below (see also \cite[Chapter III, Section 5 , p.
227]{Charatonik1979a}).

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 \g{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
$$ Y = \cl(\bigcup \{Y_n: n \in \mathbb N\})$$
is a dendrite.
See Figure A.

\FIGURE{A}{$Y_3$}

\GIF{AA}{$Y_3$}


The following properties of the dendrite $Y$ are shown in \cite[Statement
10, p. 229]{Charatonik1979a}.
\rostere
\item\label{ex-Groot-Edens} The set of all end points of $Y$ is dense in $Y$.
\item For each $n \in \mathbb N$ the set of all points of $Y$ that are
of order at least $n$ is dense in $Y$.
\item\label{ex-Groot-only1} For each natural $n \ge 3$ the dendrite $Y$ contains
exactly one point of order $n$.
\item 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 $Y$.
\item\label{ex-Groot-notsr} $Y$ is not \g{strongly rigid}.
\item\label{ex-Groot-chao} $Y$ is \gs{chaotic}{chaotic space}.
\rosteree

Recall that for any dendrite conditions~\ref{ex-Groot-Edens}
and~\ref{ex-Groot-only1} imply~\ref{ex-Groot-notsr} and~\ref{ex-Groot-chao}, see
\cite[Theorem 13, p. 22]{Charatonik1999a}. However, neither
the above mentioned conditions, nor a rough description
given in \cite{Groot+1958a}, nor the one presented in
\cite{Charatonik1979a} 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
\e{dendrite of de Groot-Wille type}
.

An example of a dendrite of de Groot-Wille type that is
\gs{chaotic}{chaotic space} but not \g{openly
chaotic} is constructed in \cite[Theorem 3.10, p.
646]{Charatonik2000a}.
