{Omiljanowski dendrite}

The
\e{Omiljanowski dendrite}
has been constructed by K. Omiljanowski in
\cite[Example 6.9, p. 182]{Charatonik1991a} (see also \cite[Remark 6.11 and Theorem 6.12,
p. 183]{Charatonik1991a}). It is defined as the closure of the union of an increasing
sequence of dendrites in the plane. We start with the unit straight line
segment denoted by $L_1$. Divide $L_1$ into three equal subsegments and in
the middle of them, $M$, locate a thrice diminished copy of the Cantor
ternary set $C$. At the midpoint of each contiguous interval $K$ to $C$
(i.e., a component $K$ of $M \setminus C$) we erect perpendicularly to $L_1$
a straight line segment whose length equals length of $K$. Denote by $L_2$
the union of $L_1$ and of all erected segments (there are countably many of
them, and their lengths tend to zero). We perform the same construction on
each of the added segments: divide such a segment into three equal parts,
locate in the middle part $M$ a copy of the Cantor set $C$ properly
diminished, at the midpoint of any component $K$ of $M \setminus C$
construct a perpendicular to $K$ segment as long as $K$ is, and denote by
$L_3$ the union of $L_2$ and of all attached segments. Continuing in this
manner we get a sequence of dendrites $L_1 \subset L_2 \subset L_3 \subset
\dots $. Finally we put
$$ L_0 = \mathrm{cl}\, (\bigcup \{L_i: i \in \Bbb N\}) $$
and call it a
\es{dendrite $L_0$}{dendrite L0}
. See Figure A.

\FIGURE{A}{dendrite $L_0$}

% PP: please, produce a picture, if possible.

The Omiljanowski dendrite $L_0$ has the following properties, \cite[Example
6.9, p. 182]{Charatonik1991a}.
\rostere
\item All ramification points of $L_0$ are of order 3.
\item The set $R(L_0)$ of all ramification points of $L_0$ is discrete
(thus nowhere dense).
\item The set of all end points of $L_0$ is nowhere dense.
\item For each maximal arc $A$ in $L_0$ the closure of the set $A \cap
R(L_0)$ contains a homeomorphic copy of the Cantor set.
\item $L_0$ is \g{monotonely equivalent} to the
\gs{dendrite $D_3$}{dendrite D3}.
\item The following conditions are equivalent for a dendrite $X$
(see \cite[Theorem 6.14, p. 185]{Charatonik1991a} and
\cite[Theorem 5.35, p. 17]{Charatonik+1994a};
compare \cite[Theorem 2.20, p. 238]{Charatonik+1998a}):
      \rostere
\item $X$ is monotonely equivalent to $D_3$;
\item $X$ is monotonely equivalent to \gs{$D_\omega$}{dendrite Domega};
\item $X$ is monotonely equivalent to \gs{$D_m$}{dendrite Dm} for each $m \in \{3, 4,
\dots, \omega\}$;
\item $X$ is monotonely equivalent to \gs{$D_S$}{dendrite DS} for each nonempty set $S
\subset \{3, 4, \dots, \omega\}$;
\item $X$ is monotonely equivalent to every dendrite $Y$ having dense
set $R(Y)$ of its ramification points;
\item $X$ is monotonely equivalent to some dendrite $Y$ having dense
set $R(Y)$ of its ramification points;
\item $X$ contains a homeomorphic copy of every dendrite $L$ such that
its set $R(L)$ of ramification points is discrete and consists of points of
order $3$ exclusively;
\item $X$ contains a homeomorphic copy of the dendrite $L_0$.
      \rosteree

According to Property~\ref{ex-other-universal-dendrites-DSmonohom}
in~\ref{ex-other-universal-dendrites} (see also
Property~\ref{ex-universal-dendrite-m-dmmonohom} in~\ref{ex-universal-dendrite-m})
any dendrite $D_S$ is \gs{monotonely homogeneous}{homogeneous
with respect to monotone mappings}. Another, less restrictive, sufficient
condition for monotone homogeneity of a dendrite is the
following (see \cite[Proposition 15, p.
364]{Charatonik+1997a}).

\item If for a dendrite $X$ the set $R(X)$ of its ramification points
is dense in $X$, then $X$ is \gs{monotonely homogeneous}
{homogeneous with respect to monotone
mappings}.

The condition $\cl R(X) = X$, being sufficient, is far from
being necessary. Namely the Omiljanowski dendrite $L_0$ has
the set $R(L_0)$ discrete, and it is \gs{monotonely homogeneous}
{homogeneous with respect to monotone
mappings}. Moreover, the following
statement holds, \cite[Proposition 20, p.
366]{Charatonik+1997a}.

\item If a dendrite $X$ contains a homeomorphic copy of $L_0$, then $X$ is
\gs{monotonely homogeneous}{homogeneous with respect to
monotone mappings}.
\rosteree

It would be interesting to know if the converse to the above statement holds
true, i.e., if containing the dendrite $L_0$ characterizes monotonely
homogeneous dendrites. In other words, we have the following question.

\textbf{Question.} Does every \gs{monotonely homogeneous}
{homogeneous
with respect to monotone mappings}
dendrite contain a homeomorphic copy of $L_0$?

The above question is closely related to a more general problem.

\textbf{Problem.} Give any structural characterization of \gs{monotonely homogeneous}
{homogeneous with respect to monotone mappings}.
