<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">{Modified Miller dendrites}

The methods used by Miller to construct his example $S$ has also been
applied to get a modification $D$ of $S$ in \cite[Chapter IV, p. 230]{Charatonik1979a}
(an outline of this construction is in \cite{Charatonik1977a}). Below we summarize
properties of this example, and recall an extension of this construction.

\rostere
\item There exists a dendrite $D$ such that:
        \rostere
\item each point of $D$ is of order nor greater than $4$;
\item for each $n \in \{1, 2, 3, 4\}$ the set of all points of $D$
which are of order $n$ is dense in $D$;
\item $D$ is \gs{strongly rigid}{strongly rigid space};
\item $D$ is \gs{chaotic}{chaotic space}.
      \rosteree
\item For any two integers $m$ and $n$ with $3 \le m &lt; n$ there exists
a dendrite $X(m,n)$ such that (see \cite[Theorem 5.5, p. 185]{Charatonik+1996a} and
compare also \cite[Theorem 27, p. 24]{Charatonik1999a}):
     \rostere
\item $\ord(x, X(m,n)) \in \{1,2,m, n\}$ for each $x \in X(m,n)$.
\item Every arc in $X(m,n)$ contains a point of order $m$ in $X(m,n)$.
\item Each of the sets: of all end points of $X(m,n)$, of all points of
order $m$ in $X(m,n)$ and of all points of order $n$ in $X(m,n)$ is dense in
$X(m,n)$.
\item $X(m,n)$ is \gs{strongly chaotic}{strongly chaotic space}.
      \rosteree

Dendrite $D$ is called a
\e{modified Miller dendrite}
.
(For a further modification of the above construction that leads to
chaotic, strongly rigid, openly rigid and not strongly chaotic dendrites,
see \cite[Theorem 3.14, p. 650]{Charatonik2000a}.)

Finally recall dendrites constructed in \cite[Examples 33 and 35, p.
28]{Charatonik1999a}.

\item For each natural $n \ge 5$, there exists a strongly rigid and not
chaotic dendrite, all points of which are of order at most $n$
(\cite[Example 33, p. 28]{Charatonik1999a}).
\item There exists a rigid dendrite which is neither chaotic nor
strongly rigid (\cite[Example 35, p. 28]{Charatonik1999a}).
\rosteree
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