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\datereceived{2019-03-28}
\dateaccepted{2020-02-24}
\dateepreuves{2020-03-02}
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\begin{document}
\frontmatter
\title{Topological properties of Wa\.zewski~dendrite groups}
\author[\initial{B.} \lastname{Duchesne}]{\firstname{Bruno} \lastname{Duchesne}}
\address{Institut Élie Cartan, UMR 7502, Université de Lorraine et CNRS\\
Boulevard des Aiguillettes, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France}
\email{bruno.duchesne@univ-lorraine.fr}
\urladdr{http://www.iecl.univ-lorraine.fr/~Bruno.Duchesne}
\thanks{The author is partially supported by projects ANR-14-CE25-0004 GAMME and ANR-16-CE40-0022-01 AGIRA}
\begin{abstract}
Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Wa\.zewski dendrite $D_\infty$ is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group $G_\infty$, we explore and prove some of its topological properties like the existence of a comeager conjugacy class, the Steinhaus property, automatic continuity, the small index subgroup property and characterization of the topology. Moreover, we describe the universal minimal flow of $G_\infty$ and of point-stabilizers. This enables us to prove that point-stabilizers in $G_\infty$ are amenable and to give a simple and completely explicit description of the universal Furstenberg boundary of $G_\infty$.
\end{abstract}
\subjclass{22F50, 57S05, 37B05}
\keywords{Wa\.zewski dendrites, groups of homeomorphisms, Polish groups, Steinhaus property, generic elements, automatic continuity, universal flows}
\altkeywords{Dendrites de Wa\.zewski, groupes d’homéomorphismes, groupes polonais, propriété de Steinhaus, éléments génériques, continuité automatique, flots universels}
\alttitle{Propriétés topologiques des groupes d’homéomorphismes des dendrites de Wa\.zewski}
\begin{altabstract}
Les groupes d’homéomorphismes des dendrites de Wa\.zewski généralisées agissent sur l’ensemble des points de branchement de la dendrite et possèdent ainsi une topologie de groupe polonais agréable. Dans cet article, nous étudions ces groupes à la lumière de cette topologie polonaise. Le groupe d’homéomorphismes de la dendrite universelle de Wa\.zewski $D_\infty$ est remarquable puisque c’est le seul avec une classe de conjugaison dense. Pour ce groupe, $G_\infty$, nous explorons et prouvons certaines de ses propriétés topologiques comme l’existence d’une classe de conjugaison comaigre, la propriété de Steinhaus, la propriété de continuité automatique, la propriété des groupes de petit indice et une caractérisation de la topologie. De plus, nous décrivons le flot minimal universel de $G_\infty$ et des stabilisateurs de points de $D_\infty$. Cela nous permet de montrer que les stabilisateurs de points de $D_\infty$ sont des groupes moyennables et de donner une description simple et explicite du bord de Furstenberg universel de $G_\infty$.
\end{altabstract}
\maketitle
\tableofcontents
\mainmatter
\Changel
\section{Introduction}
A \emph{dendrite} is a continuum (\ie a connected metrizable compact space) that is locally connected and such that any two points are connected by a unique arc (see \cite{Nadler} for background on continua and dendrites). The group of a dendrite is simply its homeomorphism group. Dendrites are tame topological spaces that appear in various domains as Berkovich projective line or Julia sets for examples. Groups acting by homeomorphisms on dendrites share some properties with groups acting by isometries on $\RR$-trees (see \eg \cite{DM_dendrites}) but some dendrite group properties are very far from properties of groups acting by isometries on $\RR$-trees, for example some have the fixed-point property for isometric actions on Hilbert spaces (the so-called property~(FH)).
In \cite{DM_dendritesII}, some structural properties of dendrite groups were studied and it was observed that two natural topologies on dendrite groups actually coincide. If $X$ is a dendrite without free arc (\ie any arc contains a branch point) then the uniform convergence on $X$ and the pointwise convergence on the set of branch points $\Br(X)$ yield the same topology on $\Homeo(X)$. Since $\Br(X)$ is countable, this yields a topological embedding
\[
\Homeo(X)\to\sinf,
\]
where $\sinf$ is the group of all permutations of the integers with its Polish topology, which is given by the pointwise convergence. The image of this embedding being closed, this means that $\Homeo(X)$ is a \emph{non-archimedean} Polish group and it becomes natural to discover which topological properties this group enjoys. For a nice survey on topological and dynamical properties of non-archimedean groups, we refer to \cite{MR3469133}.
For a non-empty subset $S\subset\overline{\NN}_{\geq3}=\{3,4,5,\dots,\infty\}$, the \emph{generalized Wa\.zewski dendrite} $D_S$ is the unique (up to homeomorphism) dendrite such that any branch point of $D_S$ has order in $S$ and for all $n\in S$, the set of points of order $n$ is arcwise-dense (\ie meets the interior of any non-trivial arc). We denote $G_S=\Homeo(D_S)$ and if $S=\{n\}$, we simply denote $D_n$ and $G_n$ for the dendrite and its group. These dendrites $D_S$ are very homogeneous, for example, the closure of any connected open subset of $D_S$ is homeomorphic to $D_S$ itself \cite[Lem.\,2.14]{DM_dendritesII}.
\subsection{Generic elements} The aim of this paper is to study some topological properties of the Polish group $G_S$ (endowed the non-archimedean topology described above). Let us first start with a proposition that separates dramatically $D_{\infty}$ from the other Wa\.zewski dendrites.
\begin{prop} The Polish group $G_S$ has a dense conjugacy class if and only if $S=\{\infty\}$.
\end{prop}
So, this shows that $G_\infty$ is remarkable among groups of Wa\.zewski dendrites and the remaining of the paper is essentially devoted to $G_\infty$.
An element in a Polish group is \emph{generic} if its conjugacy class is comeager, that is, contains a countable intersection of dense open subsets. The Polish group $G_\infty$ has generic elements. This property is sometimes called the \emph{Rokhlin property} \cite{MR2353899}.
\begin{thm}\label{ccc}The Polish group $G_\infty$ has a comeager conjugacy class.
\end{thm}
Our proof of this theorem relies on Fraïssé theory and $G_\infty$ appears as the automorphism group of some Fraïssé structure. In Section \ref{fraisse}, we detail this Fraïssé structure and some of the properties needed to prove Theorem~\ref{ccc} relying on results in \cite{MR1162490,Kechris-Rosendal}.
\subsection{Automatic continuity}A Polish group has the \emph{automatic continuity} property if any abstract group homomorphism to any separable topological group is actually continuous. This property is quite common for large Polish groups and we refer to \cite{rosendal2009automatic} for a survey. Automatic continuity is a consequence of the following property.
\begin{defn}\label{Stein}A topological group $G$ has the \emph{Steinhaus property} if there is $k\in\NN$ such that for any symmetric subset $W\subset G$ such that there is $(g_n)_{n\in\NN}$ with
\[
\bigcup_{n\in\NN}g_nW=G
\]
then $W^k$ is a neighborhood of the identity.
\end{defn}
\begin{thm}\label{144-Steinhaus}The Polish group $G_\infty$ has the Steinhaus property.
\end{thm}
In particular, we obtain the following corollary (see \cite[Prop.\,2]{Rosendal-Solecki}).
\begin{cor}\label{autcont}The Polish group $G_\infty$ has the automatic continuity property.
\end{cor}
It is well known that automatic continuity implies uniqueness of the Polish group topology. So we have another proof of a particular case of a result due to Kallman. Actually, the uniqueness of the Polish group topology for $G_S$ (with any $S\subseteq\overline{\NN}_{\geq3}$) is a direct application of \cite[Th.\,1.1]{MR831205}. So, we can speak about the Polish topology on $G_S$ without any ambiguity.
\begin{rem} The proofs use intrinsically that $\Aut(\QQ,<)$ is Steinhaus. Moreover, $G_\infty$ has the Bergman property (any isometric action on a metric space has bounded orbits) but contrary to $\Aut(\QQ,<)$ \cite[Cor.\,7]{Rosendal-Solecki}, $G_\infty$ is far to have the fixed point property for (non-necessarily continuous) actions on compact metrizable spaces. For example, the action of $G_\infty$ on the dendrite $D_\infty$ is minimal \cite{DM_dendritesII}.
\end{rem}
Corollary \ref{autcont} means that the Polish topology on $G_\infty$ is maximal among separable group topologies on this space. The following theorem goes in the other direction and shows that the Polish topology on $G_S$ is a least element among Hausdorff group topologies on $G_S$ for any $S$. See Section \ref{umin} for details.
\begin{thm}\label{tumin}For any $S\subset\overline{\NN}_{\geq3}$, the Polish group $G_S$ is universally minimal.
\end{thm}
Combining Corollary \ref{autcont} and Theorem \ref{tumin}, we obtain the following characterization of the Polish topology on $G_\infty$.
\begin{cor}There is a unique separable Hausdorff group topology on $G_\infty$.
\end{cor}
If a Polish group has \emph{ample generics} then it has the Steinhaus property. So, exhibiting ample generics is a common way to prove the Steinhaus property (see \cite[\S1.6]{Kechris-Rosendal}). In our situation, this is not possible.
\begin{prop}\label{nag} The Polish group $G_\infty$ does not have ample generics.\end{prop}
Actually, the diagonal action of $G_\infty$ on $G_\infty\times G_\infty$ by conjugation does not have any comeager orbit. Our proof relies on the same result for $\Aut(\QQ,<)$ due to Hodkinson (see \cite{MR2354899}).
\subsection{Small index property} A Polish group has the \emph{small index property} if any subgroup of small index, \ie of index less than $2^{\aleph_0}$, is open.
\begin{thm}\label{sis}The Polish group $G_\infty$ has the small index property.
\end{thm}
By definition of the topology of pointwise convergence on branch points, a basis of neighborhoods of the identity is given by pointwise stabilizers of finitely many branch points. The number of branch points being countable, these subgroups have countable index. So, Theorem \ref{sis} shows that subgroups of small index contains the pointwise stabilizer of some finite set of branch points and thus have countable index.
Let us point out that the property that subgroups of countable index are open is equivalent to the fact that any homomorphism to $\sinf$ is continuous. This last property is a particular case of the automatic continuity property.
This theorem enlightens the idea that the Polish topology on $G_\infty$ is indeed an algebraic datum: it can be recovered by subgroups of small index.
\subsection{Universal minimal flows}The group $G_\infty$ is the automorphism group of a countable structure, the set of branch points with the betweenness relation, and it is also a group of dynamical origin since it comes with its action on the compact space~$D_\infty$. So, it is natural to try to understand possible $G_\infty$-\emph{flows}, that are continuous actions of $G_\infty$ on compact spaces.
\begin{rem}The group of homeomorphisms of a metrizable compact space (endowed with the topology of uniform convergence) is separable. So, Theorem \ref{autcont} implies that any action of $G_\infty$ on a metrizable compact space by homeomorphisms is actually continuous.\end{rem}
Let $G$ be a topological group. A $G$-flow is \emph{minimal} if every orbit is dense. It is a remarkable fact there is a minimal $G$-flow which has the following universal property: Any other minimal $G$-flow is a continuous equivariant image of this largest flow. It is called the \emph{universal minimal flow} of $G$ (see for example \cite{GlasnerLNM} for details).
Usually, this universal $G$-flow is very large and not explicit at all. For $G_\infty$, we identify this universal minimal flow with a subset of the compact space of linear orders on the set of branch points. This subset consists of linear orders that are \textit{converging and convex}. They reflect the \textit{dendritic} nature of $D_\infty$. We refer to Section~\ref{umf} for definitions.
\begin{thm} The universal minimal flow of $G_\infty$ is the set of convex converging linear orders on the set of branch points of $D_\infty$.
\end{thm}
While we were working on this topic, a more general result was proved in \cite{kwiatkowska2018universal} but the description is a bit different. Our point is to show that the stabilizer of some generic converging convex linear order is actually \emph{extremely amenable}, \ie has the fixed point property on compacta. Following \cite{MR2140630}, this is equivalent to the Ramsey property for the underlying structure. For this Ramsey property, we rely on \cite{MR3436366}.
We also obtain a description of the universal minimal flow of end stabilizers and this knowledge enables us to obtain amenability results in a non-usual way. Let us observe that for a locally finite tree, the amenability of stabilizers of vertices or end points is easy but for dendrites it is not clear whether stabilizers of points are amenable in general.
\begin{thm}\label{ufb} For any point $x$ in $D_\infty$, the stabilizer of $x$ in $G_\infty$ is an amenable topological group.\end{thm}
Conversely, it is known that an amenable group acting continuously on a dendrite stabilizes a subset with at most two points \cite{ShiYe2016}.
This amenability result enables us to identify the universal Furstenberg boundary of~$G_\infty$, that is, the universal strongly proximal minimal $G_\infty$-flow. Let $\xi$ be some end point of $D_\infty$ and $G_\xi$ its stabilizer. Let us denote by $\widehat{G_\infty/G_\xi}$ the completion of $G_\infty/G_\xi$ for the uniform structure coming from the right uniform structure on $G_\infty$. We obtain a first description of the universal Furstenberg boundary.
\begin{thm}\label{ufba} The universal Furstenberg boundary of $G_\infty$ is $\widehat{G_\infty/G_\xi}$.
\end{thm}
Even if the set of end points is a dense $G_\delta$-orbit in $D_\infty$, the natural map $\widehat{G_\infty/G_\xi}\to D_\infty$ is not a homeomorphism (Proposition~\ref{nothomeo}) and thus $D_\infty$ is a Furstenberg boundary of $G_\infty$ but not the universal one. This result should be compared to the fact that the universal Furstenberg boundary of $\Homeo(\bS^1)$ is $\bS^1$ itself.
At the end of this paper, we give another description of this universal Furstenberg boundary. It appears as a closed subset of a natural countable product of totally disconnected compact spaces. The description is simple and shows that it is a countable collection of $G_\infty$-orbits. This universal Furstenberg boundary is the space $K$ that appears in Subsection~\ref{isomorphic}.
\subsubsection*{Acknowledgements}
This paper greatly benefited from the participation to the conference \textit{Geometry and Structure of Polish Groups} held in Casa Mathemática Oaxaca, Mexico in June 2017. The author thanks the organizers for the invitation and people he had pleasure to speak with and who helped to improve this paper: E.\,Glasner, A.\,Kwiatkowska, F.\,Le Maître, J.\,Melleray, L.\,Nguyen Van Thé, C.\,Rosendal, T.\,Tsankov and P.\,Wesolek. This paper would not have existed without previous works with N.\,Monod. It is pleasure to thank him for insightful discussions.
\section{Wa\.zewski dendrites and their homeomorphism groups}\label{wazewski}
\subsection{Wa\.zewski dendrites} A \emph{dendrite} is a connected metrizable compact space that is locally connected and such that any two points $x,y$ are connected by a unique arc $[x,y]$. Simple examples are given by compactifications of locally finite simplicial trees. Some examples are more complicated. For example, the Julia set of the polynomial map $z\mto z^2+i$ of the complex line $\CC$ is a dendrite (See Figure \ref{Julia}). A \emph{subdendrite} is a closed and connected subset $S$ of a dendrite $D$. It is a dendrite on its own and there is a retraction $\pi_S\colon D\to S$ such that for any $x\in D$ and $y\in S$, $\pi_S(x)\in[x,y]$. This retraction is also called the \emph{first-point map} to $S$.
\begin{figure}[b]
\includegraphics[width=.4\textwidth]{dendritejulia.png}
\caption{The Julia set of $z\mto z^2+i$. Picture realized with \textit{Mathematica}.}\label{Julia}
\end{figure}
In a dendrite $X$, there are three types of points $x\in X$, according to the cardinal of~$\cC_x$, the space of connected components of $X\setminus\{x\}$. This number is at most countable and is called the \emph{order} of $x$.
\begin{itemize}
\item If the complement $X\setminus\{x\}$ remains connected, $x$ belongs to the set $\Ends(X)$ of \emph{end points}.
\item If $x$ separates $X$ into two components, it is a \emph{regular point}.
\item Otherwise, it is at least 3 and $x$ belongs to the set $\Br(X)$ of \emph{branch points}.
\end{itemize}
Let $X$ be a dendrite and $c\colon X^3\to X$ be the center map, that is, $c(x,y,z)=[x,y]\cap[y,z]\cap[z,x]$, which is reduced to a unique point. Let us make a few observations:
\begin{itemize}
\item $c$ is symmetric,
\item $c(x,y,z)=z$ if and only if $z\in[x,y]$,
\item if $c(x,y,z)\notin\{x,y,z\}$, $c(x,y,z)$ is a branch point
\item and in particular, $\Br(X)$ is $c$-invariant, \ie $c(\Br(X))=\Br(X)$.
\end{itemize}
In Bowditch's terminology, $(X,c)$ is a median space \cite{BowditchAMS}. For two points $x\neq y$ in a dendrite $X$, we denote by $D(x,y)$ the connected component of $X\setminus\{x,y\}$ that contains $]x,y[$. Let us say that a subset $Y$ of $X$ is $c$-\emph{closed} if for any $x,y,z\in Y$, $c(x,y,z)\in Y$. For $Y\subset X$, we define the $c$-\emph{closure} of $Y$ to be $c(Y^3)$, which happens to be the smallest $c$-closed subset of $X$ containing $Y$.
\begin{lem} For any $Y\subset X$, $c(Y^3)$ is $c$-closed.
\end{lem}
\begin{proof} Let $x_1,x_2,x_3\in c(Y^3)$ and $m=c(x_1,x_2,x_3)$. If $m\in\{x_1,x_2,x_3\}$ then we are done. Otherwise, for each $i=1,2,3$, one can find a point $y_i\in Y$ that is not in $C_{x_i}(m)$, the connected component of $X\setminus\{x_i\}$ that contains $m$. Thus $m=c(y_1,y_2,y_3)\in c(Y^3)$.\end{proof}
Let $S$ be a non-empty subset of $\overline{\NN}_{\geq3}=\{3,4,5,\dots,\infty\}$. The \emph{Wa\.zewski dendrite}~ $D_S$ is the unique (up to homeomorphism) dendrite such that all orders of branch points belong to $S$ and for any $n\in S$, the set of points of orders $n$ is arcwise-dense. See \cite[\S12]{DM_dendrites} for a few historical references and a reference to the proof of this characterization. With this characterization, it easy to see that the closure of any open connected subset in $D_S$ is actually homeomorphic to $D_S$. We denote by $G_S$ the homeomorphism group of $D_S$. If $S=\{n\}$, we simply denote $D_n$ and $G_n$ for the dendrite and its group. For example, $D_\infty$ appears to be homeomorphic to the Berkovich projective line over $\CC_p$. See \cite[Fig.\,1] {Hrushovski-Loeser-Poonen} for explanations and a nice picture of this dendrite.
Let us recall some properties of the groups $G_S=\Homeo(D_S)$ proved in \cite[\S\S 6\,\&\,7]{DM_dendritesII}. The first one shows how homogeneous the dendrite $D_S$ is.
To any finite subset $F$ of the dendrite $D_S$, we associate a finite vertex-labeled simplicial tree $\langle F \rangle$ as follows. The sub-dendrite $[F]$, \ie the smallest subdendrite containing $F$, is a finite tree in the topological sense, \ie the topological realization of a finite simplicial tree. Such a simplicial tree is not unique because degree-two vertices can be added or removed without changing the topological realization. We choose for $\langle F \rangle$ to retain precisely one degree-two vertex for each element of $F$ which is a regular point of the dendrite $[F]$. Thus, $\langle F \rangle$ is a tree whose vertex set contains $F$. Finally, we label the vertices of $\langle F\rangle$ by assigning to each vertex its order in $D_S$.
\begin{prop}[{\cite[Prop.\,6.1]{DM_dendritesII}}]\label{prop:extension}
Given two finite subsets $F, F'\se D_S$, any isomorphism of labeled graphs $\langle F \rangle\to \langle F' \rangle$ can be extended to a homeomorphism of~$D_S$.
\end{prop}
This simple proposition has strong corollaries for $G_S$. For example, $G_S$ acts $2$\nobreakdash-transitively on the set of points of a given order. Moreover, $G_S$ is a simple group and if $S$ is finite then the action of $G_S$ on the set of branch points (which is countable) is oligomorphic \cite[Cor.\,6.7]{DM_dendritesII}. For an introduction to oligomorphic groups, we~refer to~\cite{MR2581750}.
The structure of the group $G_S$ completely determines the dendrite $D_S$: if $G_S$ and~$G_{S'}$ are isomorphic then $S=S'$ and an automorphism of $G_S$ is always given by a conjugation \cite[Cor.\,7.4 \& 7.5]{DM_dendritesII}.
Since the action on the set of branch points $\Br(D_S)$ completely determines the action, $G_S$ embeds as a closed subgroup of $\sinf$. With this topology, if $S$ is finite then the Polish group $G_S$ has the strong Kazhdan property (T) \cite[Cor.\,6.9]{DM_dendritesII}. Without the assumption of finiteness of $S$, the discrete group $G_S$ has Property (OB) (every action by isometries on a metric space has bounded orbits) \cite[Cor.\,6.12]{DM_dendritesII}.
We will need to construct global homeomorphisms from patches of partial homeomorphisms. This is possible thanks to the following lemma \cite[Lem.\,2.9]{DM_dendritesII}.
\begin{lem}[Patchwork lemma]\label{lem:patchwork}
Let $\sU$ be a family of disjoint open connected subsets of a dendrite $X$ and let $(f_U)_{U\in \sU}$ be a family of homeomorphisms $f_U \in\Homeo(U)$ for $U\in\sU$. Suppose that each $f_U$ can be extended continuously to the closure $\overline U$ by the identity on the boundary $\overline U\setminus U$.
Then the map $f\colon X\to X$ given by $f_U$ on each $U\in \sU$ and the identity everywhere else is a homeomorphism.
\end{lem}
\subsection{Dynamics of individual elements}
Let $g$ be a homeomorphism of a dendrite~$X$. An arc $[x,y]\subset X$ is \emph{austro-boreal} for $g$ if $x\neq y$ are fixed and there is no fixed-point in $]x,y[$. Observe that in this case, the restriction of the action of $g$ on $]x,y[$ is conjugated to an action by a non-trivial translation on the real line.
For a non-trivial arc $[x,y]$, we denote by $D(x,y)$ or $D([x,y])$ the connected component of $X\setminus\{x,y\}$ that contains $]x,y[$. We denote by $D(g)$ the union of all $D(I)$, where $I$ is an austro-boreal arc for $g$ and by $K(g)$ its complement in $X$.
The following proposition \cite[Prop.\,10.6]{DM_dendrites} describes the dynamics of a homeomorphism of a dendrite.
\begin{prop}\label{prop:indiv}
The decomposition $X=D(g) \sqcup K(g)$ has the following properties.
\begin{enumeratei}
\item\label{prop:indivi} $D(g)$ is a (possibly empty) open $g$-invariant set on which $g$ acts properly discontinuously. In particular, $K(g)$ is a non-empty compact $g$-invariant set.\label{pt:tectonic:dec}
\item\label{prop:indivii} $K(g)$ is a disjoint union of subdendrites of $X$. Moreover, $g$ preserves each such subdendrite and has a connected fixed-point set in each.\label{pt:tectonic:K}
\end{enumeratei}
\end{prop}
The subdendrites that appear in \eqref{prop:indivii} in the above proposition are actually the connected component of $K(g)$. Let us precise the action of $g$ on these connected components.
\begin{lem}\label{rem:kg}If $C$ is a connected component of $K(g)$, then $g$ permutes the connected components of $C\setminus\Fix(g)$, where $\Fix(g)$ is the set of fixed points of $g$. Moreover, none of these connected components of $C\setminus\Fix(g)$ is invariant.
\end{lem}
\begin{proof} The connected component $C$, which is a subdendrite, is $g$-invariant by Proposition~\ref{prop:indiv} and contains at least one fixed point by the fixed point property for dendrites (See for example \cite[Lem.\,2.5]{DM_dendrites}). So $g$ permutes the connected components of $C\setminus\Fix(g)$. Let $C'$ be a connected component of
$C\setminus\Fix(g)$. Its closure $C'$ is a subdendrite. Let $x\in C'$ and $y\in\overline{C'}$ be distinct points. There is a sequence $(x_n)$ converging to $y$ such that $x_n\in C'$ for all $n\in\NN$. Since $C'$ is connected $[x,x_n]\subset C'$ for all $n\in\NN$. For all $z\in{}]x,y[$ and $n$ large enough, $z\in [x,x_n]$. Thus $[x,y[{}\subset C'$. For $y' \in\overline{C'}\setminus C'$ such that $y'\neq y$, the point $c(x,y,y')\in C'$ separates $y$ and $y'$. Since $\Fix(g)\cap C$ is connected, $\overline{C'}$ contains exactly one fixed point. This fixed point is an end point of the subdendrite $\overline{C'}$ because $C'$ is connected. Assume $C'$ is $g$-invariant. Then by \cite[Lem.\,4.8]{DM_dendritesII}, if $g$ fixes an end point then $g$ has at least two fixed points. So, $g$ has two fixed points in $\overline{C'}$ and we have a contradiction. \end{proof}
Let $X$ be a dendrite and $g\in\Homeo(X)$. It will be useful for us to decide where a point $x\in X$ lies in the decomposition $X=K(g)\sqcup D(g)$ from Proposition \ref{prop:indiv}, using only finitely many points in the $g$-orbit of $x$.
\begin{lem}\label{lem:c}Let $g\in\Homeo(X)$ and $x\in X$. Then,
\begin{itemize}
\item $x$ is in the interior of some austro-boreal arc if and only if $g(x)\in{}]x,g^2(x)[$,
\item $x\in D(g)$ if and only if
\[
c(g(x),g^2(x),g^3(x))\in{}]c(x,g(x),g^2(x)),c(g^2(x),g^3(x),g^4(x)[,
\]
\item $x\in K(g)$ if and only if $[x,g(x)]\cap\Fix(g)\neq\emptyset$.
\end{itemize}
\end{lem}
\begin{proof} Let us start with elements in the interior of some austro-boreal arc. Let $[y,z]$ be some austro-boreal arc for $g$. If $x\in{}]y,z[$ then $g(x)\in{}]x,g^2(x)[$ because the action of $g$ on $]y,z[$ is conjugated to an action on the real line by translation. Conversely, if $g(x)\in{}]x,g^2(x)[$ then for any $n\in\ZZ$, $g^n(x)\in{}]g^{n-1}(x),g^{n+1}(x)[$ and thus $\overline{\bigcup_{n\in\ZZ}[g^n(x),g^{n+1}(x)]}$ is an austro-boreal arc.
If $x\in D(y,z)$ for some austro-boreal arc $[y,z]$, let $p$ be the image of $x$ under the first-point map to $[y,z]$. The point $p$ lies in $]y,z[$, so $gp=c(x,g(x),g^2(x))$ belongs to $]y,z[$ and satisfies the first item of the lemma, and thus
\[
gc(x,g(x),g^2(x))\in[c(x,g(x),g^2(x)),g^2c(x,g(x),g^2(x))]
\]
by equivariance of the map $c$. That is, $c(g(x),g^2(x),g^3(x))$ belongs to
\[
]c(x,g(x),g^2(x)),c(g^2(x),g^3(x),g^4(x)[\,.
\]
Conversely, if
\[
c(g(x),g^2(x),g^3(x))\in{}]c(x,g(x),g^2(x)),c(g^2(x),g^3(x),g^4(x)[,
\]
then by the first part this means that $c(g(x),g^2(x),g^3(x))$ lies in some austro-boreal arc. Let $y,z$ be the ends of this arc. Then, by construction, $x\in D(y,z)$.
Let $x\in K(g)$, let $C$ be its connected component in $K(g)$ and let $C_0$ be its connected component in $C\setminus\Fix(g)$. By Lemma~\ref{rem:kg}, $gC_0\cap C_0=\emptyset$ and thus there is a fixed point~$p$ in $[x,g(x)]$. Conversely, if there is a fixed point $p$ in $[x,g(x)]$ then $x\notin D(g)$ since the action of $g$ on $D(g)$ is properly discontinuous.
\end{proof}
The decomposition $X=K(g)\sqcup D(g)$ is not really a group invariant for the cyclic group $\langle g\rangle$ generated by $g$. Each part is $\langle g\rangle$-invariant but the decomposition is not the same for every element of $\langle g\rangle$. Let us illustrate this phenomenon.
\begin{example} Let us fix $\xi_\pm$ two end points of the Wa\.zewski dendrite $D_3$. Let $x$ be some regular point of $D_3$ and let $C_1,C_2$ be the two connected components of $D_3\setminus\{x\}$. Let $\varphi_i$ be an homeomorphism from $D_3\setminus\{\xi_+\}$ to $C_i$. Let $\gamma$ be some homeomorphism of $D_3$ such that $[\xi_-,\xi_+]$ is austro-boreal for $\gamma$. We define an homeomorphism $g$ of $D_3$ fixing $x$ and such that $g|_{C_1}=\varphi_2\circ\varphi_1^{-1}$ and $g|_{C_2}=\varphi_1\circ\gamma\circ\varphi_2^{-1}$. The map $g$ is well-defined thanks to Lemma~\ref{lem:patchwork}. By construction, we have $K(g)=D_3$ and $D(g)=\emptyset\ $ but $K(g^2)=\{x\}$ and $D(g^2)=C_1\cup C_2$. \end{example}
Nonetheless, we have the following inclusions.
\begin{lem}\label{KD}Let $X$ be a dendrite and $g\!\in\!\Homeo(X)$. For any $n\!\in\!\NN$, $D(g)\!\subset\!D(g^n)$ and $K(g^n)\subset K(g)$.
\end{lem}
\begin{proof} It suffices to prove the first inclusion and pass to the complement to get the other one. If an arc is austro-boreal for $g$, it is austro-boreal for any of its non-trivial power and thus $D(g)\subset D(g^n)$.
\end{proof}
\section{Fraïssé theory and generic elements}\label{fraisse} We use the notations of \cite{Kechris-Rosendal} and denote by $\bK$ the Fraïssé structure associated to the action of $G_S$ on the countable set $\Br(D_S)$ (see \cite[\S 1.2]{Kechris-Rosendal}) and by $\cK$ the Fraïssé class of finite substructures of $\bK$. In particular, $\bK$ is the Fraïssé limit of $\cK$ and $\Aut(\bK)=G_S$. Let us briefly explain what is this structure. The structure $\bK$ is the set $\Br(D_S)$ with the all relations $R_{i,n}\subset \Br(D_S)^n$ given by orbits of the diagonal actions of $G_S$ on $\Br(D_S)^n$.
Let us briefly recall what it means to be a Fraïssé class. The class $\cK$ is a countable class of finite structures over some fixed countable signature that enjoys the following properties:
\begin{enumerate}
\item\textit{Hereditary property.} For any $\bB\!\in\!\cK$ and $\bA\!\leq\!\bB$ (\ie $\bA$ can be embedded in~$\bB$), $\bA\in\cK$.
\item\textit{Joint embedding property.} For any $\bA,\bB\in\cK$, there is $\bC$ such that $\bA,\bB\leq\bC$.
\item\textit{Amalgamation property. }For $\bA,\bB,\bC\in\cK$, if $f\colon\bA\to\bB$ and $g\colon\bA\to\bC$ are embeddings then there is $\bD\in\cK$ and embeddings $r\colon\bB\to\bD,\ s\colon\bC\to\bD$ such that $r\circ f=s\circ g$.
\end{enumerate}
The structure $\bK$, the Fraïssé limit of $\cK$, is the countable structure over the same signature such that any finite substructure belongs to $\cK$ and $\bK$ is ultra-homogeneous: any partial isomorphism between finite substructures extends to a global isomorphism.
\begin{rem}\label{structure} The structure $\bK$ is, a priori, given by $\Br(D_S)$ (as underlying set) and infinitely many relations corresponding to orbits in $\Br(D_S)^n$ for $n\in\NN$. But actually, $G_S$ can be realized as the automorphism group of a structure given by $\Br(D_S)$ and a unique relation: the betweenness relation $B$, where $B(z;x,y)\Leftrightarrow z\in[x,y]$. A~bijection of $\Br(D_S)$ that preserves the betweenness relation is actually given by a homeomorphism of $D_S$ \cite[Prop.\,2.4]{DMW}.
A betweenness relation $B$ is of \emph{positive type} if for any $x,y,z$ there is $w$ such that $B(w;x,y)\wedge B(w;y,z)\wedge B(w;z,x)$. Moreover, a finite set with a betweenness relation with positive type (as it is the case for finite subsets of $\Br(D_\infty)$ closed under the center map) has a tree structure \cite[Lem.\,29.1]{MR1388893} and thus embeds in the set of branch points of the universal dendrite $D_\infty$. We refer to \cite{MR1388893} for details about betweenness relations and being of positive type. By Proposition \ref{prop:extension}, any isomorphism between two finite subtree of $\Br(D_\infty)$ can be realized as the restriction of some element of~$G_\infty$. This way $\Br(D_\infty)$ with the betweenness relation is the Fraïssé limit of the class of finite betweenness structures with positive type.
Let us observe that the center map can be defined only in terms of the betweenness relation. Actually, $c(x,y,z)=w$ is equivalent to $B(w;x,y)\wedge B(w;y,z)\wedge B(w;z,x)$.\end{rem}
We also denote by $\cK_p$ the class of systems $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$, where $\bB,\bC\subseteq \bA$ are finite substructures of $\bK$ and $\varphi$ is an isomorphism between these substructures. Let $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$ and $\cT=\langle \bD,\psi\colon\bE\to\bF\rangle$ be two systems of $\cK_p$. An \emph{embedding} of $\cS$ into $\cT$ is an embedding of structures $f\colon \bA\to\bD$ that induces an embedding of $\bB$ in $\bE$, an embedding of $\bC$ in $\bF$ and such that $f\circ\varphi\subseteq \psi\circ f$. In that case, we also say that $\cT$ is an \emph{extension} of $\cS$. This notion of embeddings allows us to speak about the joint embedding property (JEP) or the amalgamation property (AP) for $\cK_p$. A subclass $\cL$ of $\cK_p$ is \emph{cofinal} if for any system $\cS\in\cK_p$, there is $\cT\in\cL$ and an embedding of $\cS$ into $\cT$.
For a system $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$ and $g\in G_S$, we say that $g$ \emph{induces} $\varphi$ if there is $A\subset \Br(D_S)$ and an isomorphism $f\colon \bA\to A$ such that $\varphi=f^{-1}gf$. In this case, by an abuse of notation, we consider $\bA$ as a subset of $\Br(D_S)$ and forget about $f$.
\subsection{Existence of a dense conjugacy class}
The following proposition shows that $G_\infty$ is remarkable among all the Wa\.zewski groups.
\begin{prop}\label{prop:jep} The Polish group $G_S$ has a dense conjugacy class if and only if $S=\{\infty\}$.
\end{prop}
\begin{proof} Thanks to \cite[Th.\,1.1]{Kechris-Rosendal}, it suffices to show that $\cK_p$ satisfies (or not) the joint embedding property.
Assume that $S$ contains $n\neq\infty$. Choose a point $x\in D_S$ of order $n$ and $x_1,\dots,x_n$ in distinct connected components of $D_S\setminus\{x\}$ such that there exists $g\in G_S$ with $gx_i=x_{i+1}$ ($i\in\ZZ/n\ZZ$). We set $\bA=\bB=\bC=\{x,x_1,\dots,x_n\}$, $\varphi$ to be the restriction of $g$ on $\bB$ and $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$. Let $\cT=\langle \bD,\psi\colon\bE\to\bF\rangle$, where $\bD=\bE=\bF$ are two points and $\psi$ is the identity. Now, $\cS$ and $\cT$ do not have a joint embedding because any extension of $\varphi$ in $\Homeo(D_S)$ has a unique fixed point, namely $x$.
Now, assume $S=\{\infty\}$. We claim that a partial isomorphism between finite substructures can be extended to a homeomorphism of $D_\infty$ fixing a branch point and fixing pointwise a connected component of the complement of this fixed point.
Assume the claim holds true. Let $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$ and $\cT=\langle \bD,\psi\colon\bE\to\bF\rangle$ be elements of $\cK_p$. Thanks to the claim, we assume that $\varphi$ is induced by $f\in G_\infty$ that fixes a point $x\in\Br(D_\infty)$ and $\psi$ is induced by $g\in G_\infty$ that fixes a point $y\in\Br(D_\infty)$. Moreover, conjugating $g$ with some element of $G_\infty$ if necessary, we may assume that~$y$ (\resp $x$) is in a component of $\cC_x$ (\resp $\cC_y$) pointwise fixed by $f$ (\resp $g$). Now, define~$h$ to be the identity on $D(x,y)$, acts like $f$ on the support of $f$ and like $g$ on the support of $g$. This $h$ yields a joint extension of $\varphi$ and $\psi$.
Let us prove the claim. Any $g\in G_\infty$ has a fixed point $x\in D_\infty$ and thus permutes the components of $D_\infty\setminus\{x\}$. These components are all homeomorphic to $D_\infty\setminus\{\xi\}$, where $\xi$ is some end point. If $x$ is a branch point, we may glue a new copy $D$ of $D_\infty$ by identifying some end point in $D$ with $x$. If $x$ is not a branch point, we glue countably many copies of $D_\infty$. The new dendrite is $D_\infty$ once again and $x$ is a branch point. One obtains a new dendrite homeomorphic to $D_\infty$ and one can extend $g$ by the identity on the new copies of $D_\infty$.\end{proof}
\subsection{Existence of a comeager conjugacy class} Let us recall that for a dendrite $X$ and two points $x,y\in X$, we denote by $D(x,y)$ the connected component of $X\setminus\{x,y\}$ that contains $]x,y[$.
For the remaining of this section, we consider only the dendrite $D_\infty$ and its associated Fraïssé class $\cK$. In \cite{MR1162490}, Truss introduced a general way to prove existence of generic elements in automorphism groups of countable structures. To prove this existence, it suffices to show that $\cK_p$ has the joint embedding property (JEP) and the amalgamation property (AP) defined above. Actually, a cofinal version of (AP) is sufficient. In \cite{Kechris-Rosendal}, a weaker condition, the weak amalgamation property (WAP) has been shown to be the necessary and sufficient amalgamation condition.
\begin{rem}\label{notAP} The class $\cK_p$ does not have (AP). Let us consider the simple example $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$, where $x,y$ are two distinct points of $\Br(D_\infty)$, $B=\{x\}$, $C=\{y\}$, $A=\{x,y\}$ and $\varphi(x)=y$. Actually, $\varphi$ can be realized by an automorphism~$g_1$ that fixes a point $p$ in $[x,y[$ or by an element $g_2$ such that $[x,y]$ is included in some austro-boreal arc for $g_2$. If $\varphi$ is extended by $\varphi_1$ the restriction of $
g_1$ on $\{x,p\}$ and by~$\varphi_2$ the restriction of $g_2$ on $\{x,y\}$, it is not possible to amalgamate~$\varphi_1$ and~$\varphi_2$ over~$\varphi$. Actually, if $\psi$ is an amalgamation, it is given by an element $g\in G_\infty$ that has a fixed point in $[x,y]$ because of $\varphi_1$ and simultaneously such that $[x,y]$ is included in some austro-boreal arc for $g$ because of the first point in Lemma \ref{lem:c}. Thus we have a contradiction.
\end{rem}
Below, we define a subclass $\cL$ of $\cK_p$ for which we show cofinality and the amalgamation property. As explained in Remark~\ref{structure}, we consider the structure $\bK$ with the betweenness relation and the associated center map $c$ and for a finite structure of positive type $\bA\in\cK$ (that is, a $c$-closed subset of $\bK$) and points $x,y,z\in\bA$, we write $x\in[y,z]$ if $c(x,y,z)=x$ that is, $B(x;y,z)$. For $x,y\in\bA$, we define $D(x,y)=\{z\in \bA, c(x,y,z)\notin\{x,y\}\}$. Let us observe that if $\bA$ is embedded in $\Br(D_\infty)$ then these definitions are consistent with the ones in $D_\infty$. For a system $\langle \bA,\varphi\colon\bB\to\bC\rangle\in\cK_p$ and $x\in \bB$, we write $\varphi^n(x)$ for $n\in\NN$ if $\varphi(x),\dots,\varphi^{n-1}(x)\in\bB$ and define $\varphi^n(x)$ to be $\varphi(\varphi^{n-1}(x))$. In particular, when this notation is used it implies implicitly that $\varphi(x),\dots,\varphi^{n-1}(x)$ are well-defined and belong to $\bB$. If there is $n\in \NN$ such that $\varphi^n(x)=x$ then we say that $x$ is $\varphi$-\emph{periodic}. In that case, its \emph{period} is $\inf\{n>0,\ \varphi^n(x)=x\}$.
Let $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle\in\cK_p$ be a system. We define a $\varphi$-\emph{orbit} to be an equivalence class under the equivalence relation on $\bB\cup\bC$ generated by $x\sim_\varphi y\ssi y=\varphi(x)$ or $x=\varphi(y)$.
\begin{defn}\label{class}We consider the subclass $\cL\!\subseteq\!\cK_p$ of systems
\[
\cS=\langle \bA,\varphi\colon\bB\!\to\!\bC\rangle\in\cK_p
\]
with $\bA,\bB$ and $\bC$ of positive type and that satisfy the following conditions. There exists $\bB_0\subset \bB$ such that
\begin{enumerate}
\item\label{class1} for any $y\in\bB$, there is an $x\in\bB_0$ and $k$ non-negative integer such that $y=\nobreak\varphi^k(x)$.
\item\label{class2} For any $x\in \bB_0$, $x$ is $\varphi$-periodic or there exists $n\in \NN$ such that
\[
c\left(\varphi^n(x),\varphi^{2n}(x),\varphi^{3n}(x)\right)\in\left]c\left(x,\varphi^n(x),\varphi^{2n}(x)\right),c\left(\varphi^{2n}(x),\varphi^{3n}(x),\varphi^{4n}(x)\right)\right[.
\]
\item\label{class3} For any $x\in \bB$ such that $x$ is not $\varphi$-periodic, there exist $\varphi$-periodic points $y,z\in \bB$ such that $x\in D(y,z)$.
\item\label{class4} For any $x,y\in\bB_0$ such that there exist $n,m\in\NN$ with $\varphi^n(x)\in{}]x,\varphi^{2n}(x)[$ and $\varphi^m(y)\in{}]y,\varphi^{2m}(y)[$.
\begin{itemize}
\item if the $\varphi^n$-orbit of $x$ and the $\varphi^m$-orbit of $y$ are separated (\ie no point of one of the orbit is between two points of the other) then there exists a $\varphi$-periodic point $z\in\bB$ such that $z\in [x,y]$,
\item in the other case there exist $x_0,y_0\in \bB_0$ and $k\in\NN$ such that
$\sbullet$ the $\varphi$-orbit of $x$ is $\{x_0,\dots,\varphi^{k}(x_0)\}$, the $\varphi$-orbit of $y$ is $\{y_0,\dots,\varphi^{k}(y_0)\}$,
$\sbullet$ $y_0\in[x_0,\varphi^l(x_0)]$ or $x_0\in[y_0,\varphi^l(y_0)]$, where $l$ is the minimum such that $\varphi^l(x_0)\in{}]x_0,\varphi^{2l}(x_0)[$ or $\varphi^l(y_0)\in{}]y_0,\varphi^{2l}(y_0)[$
$\sbullet$ and $k$ is a multiple of $l$.
\end{itemize}
\item\label{class5} If $x\in\bB$ and $y,z\in \bB$ are $\varphi$-periodic points such that $x\in D(y,z)$ then the size of the $\varphi$-orbits of $x$ and of $c(x,y,z)$ are the same.
\item\label{class6} The set $\bA$ is the $c$-closure of $\bB$ and $\bC$. That is, for any $x\in \bA$, there exist $x_1,x_2,x_3\in \bB\cup\bC$ such that $x=c(x_1,x_2,x_3)$.
\end{enumerate}
\end{defn}
Let us explain this definition. The first point means that there is some initial set $\bB_0$ such that any point of $\bB$ lies in some positive $\varphi$-orbit of $\bB_0$. For the second point, it means that any point of $\bB_0$ is a fixed point of $g^n$ or lies in $D(g^n)$ for any $g\in G_\infty$ that induces $\varphi$ (Lemma~\ref{lem:c}). This removes the indeterminacy that appears in Remark~\ref{notAP}. More precisely, these conditions imply that no extension of such a system can merge distinct $\varphi$-orbits (Lemma~\ref{atmostoneorbit}). The third point means that if $x$ belongs to $D(g^n)$, where $g$ induces $\varphi$, then it lies in the connected component between two fixed points of $g^n$. Condition \eqref{class4} means that if $x,y$ lie in some austro-boreal part for some power of $g$ inducing $\varphi$ and the orbits under these powers do not intertwine, then they are separated by some periodic point.
\begin{lem}\label{fixpt} The class of systems $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle\in\cK_p$ such that there is $x_0\in \bB$ with $\varphi(x_0)=x_0$ is cofinal in $\cK_p$.
\end{lem}
\begin{proof}Since $\Br(D_\infty)$ is a Fraïssé limit, we know that for any system $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle$, we may identify $\bA$ with a subset of $\Br(D_\infty)$ and $\varphi$ with some restriction to $\bA$ of an automorphism $g\in\Homeo(D_\infty)$. Since dendrites have the fixed point property, $g$ has a fixed point $x$ in $D_\infty$. If this point $x$ is a branch point, it suffices to add it to $\bA,\bB$ and $\bC$ (and possibly the finitely many points $c(x,y,z)$ for $y,z\in \bA,\bB$ or $\bC$) and to extend $\varphi$ with $\varphi(x)=x$ (or by the value of $g$ on the points $c(x,y,z)$ for $y,z\in\bB$). If $x$ is not a branch point then we can reduce to the situation where $x$ is a fixed branch point by the following construction. We glue infinitely many copies of $D_\infty$ to $x$ by identifying a branch point of each copy with $x$. We extend $g$ on the new branches around $x$ by the identity (which is possible thanks to the patchwork lemma). The new dendrite is homeomorphic to $D_\infty$ once again and we are back in the situation where $g$ has a fixed branch point.
\end{proof}
Let $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle\in\cK_p$ be a system with a fixed point $x_0\in\bB$. We define a \emph{branch} around $x_0$ to be an equivalence class in $\bA\setminus\{x_0\}$ under the relation $x\sim_{x_0} y \ssi \neg B(x_0;x,y)$.
For two branches around $x_0$, we write $D_1\sim_\varphi D_2$ if there is $x\in D_1\cap \bB$ such that $\varphi(x)\in D_2$. Observe that for another $y\in \bB\cap D_1$ then $\varphi(y)\in D_2$ because $\varphi$ preserves the betweenness relation. In that case, we write $\varphi(D_1)=D_2$ even if this equality is not true for the underlying sets (we only have $\varphi(D_1\cap \bB)\subset D_2$). We also take the liberty to write recursively $\varphi^n(D_1)$ for $\varphi(\varphi^{n-1}(D_1))$ if $\varphi^{n-1}{(D_1)}\cap \bB\neq\emptyset$. We still denote by $\sim_\varphi$ the equivalence relation generated by this relation. A $\varphi$-\emph{orbit of branches} is an equivalence class of branches under this equivalence relation.
\begin{lem}\label{phi-orbit} For any $\varphi$-orbit of branches $E$ around $x_0$, there is a branch $D$ and $n\in\NN$ with $D\cap \bB\neq\emptyset$ such that $E=\{D,\varphi(D),\dots,\varphi^{n-1}(D)\}$.
\end{lem}
\begin{proof} Let us first prove that if $D$ and $D'$ are two branches around $x_0$ such that $\varphi(D)=\varphi(D')$ then $D=D'$. In fact, if $x\in D$, $y\in D'$ with $\neg B(x_0;\varphi(x),\varphi(y))$ then $\neg B(x_0;x,y)$ and thus $D=D'$.
If $D,D'$ are in $E$ then there is chain $D_0,\dots,D_k$ such that $D_0=D$, $D_k=D'$ and $\varphi(D_i)=D_{i+1}$ or $\varphi(D_{i+1})=D_i$ for each $i=0,\dots,k-1$. One shows by induction on the length of the chain that $D=\varphi^k(D')$ or $D'=\varphi^k(D)$.
Now, let $\{D,\varphi(D),\dots,\varphi^{n-1}(D)\}$ be a maximal such chain with distinct elements (which exists since $E$ is finite). Let $D'\in E$. Then there is a minimal $k\in\NN$ such that
\[
\varphi^k(D)=D'\quad\text{or}\quad \varphi^k(D')=D.
\]
In the first case, by maximality of $\{D,\varphi(D),\dots,\varphi^{n-1}(D)\}$,
\[
k\leq n-1\quad\text{and}\quad D'\in\{D,\varphi(D),\dots,\varphi^{n-1}(D)\}.
\]
In the second case, by maximality again, one has
\[
\{D',\varphi(D'),\dots,\varphi^{n-1+k}(D')\}=\{D,\dots,\varphi^{n-1}(D)\}.
\]
Thus $E=\{D,\varphi(D),\dots,\varphi^{n-1}(D)\}$.
\end{proof}
Observe that in Lemma~\ref{phi-orbit}, it is possible that $\varphi^{n-1}(D)\cap\bB\neq\emptyset$ and $\varphi^n(D)=D$.
Let $E$ be a $\varphi$-orbit of branches and $\bE$ the union of its branches (which is $c$-closed). We define $\cS_E=\langle \bA_E,\varphi_E\colon \bB_E\to \bC_E\rangle$, where $\bA_E=(\bA\cap \bE)\cup\{x_0\}$, $\bB_E=\bB\cap\bA_E$, $\bC_E=\bC\cap\bA_E$ and $\varphi_E$ is the restriction of $\varphi$ to $\bB_E$. Observe
that $\cS_E\in\cK_p$ and if $\cS\in\cL$ then $\cS\in\cL$ as well.
\begin{lem}\label{lem:addfixedpoint} Let $\cS=\langle \bA,\varphi\colon \bB\to\bC\rangle\in \cK_p$ be a system with a fixed point $x_0$ and points $x_1,\dots,x_k\in\bA$ such that $x_1,\dots,x_{k-1}\in\bB$, $x_1\notin \bC$, $[x_{i+1},x_0]\subset [x_i,x_0]$ and $\varphi(x_i)=x_{i+1}$ for all $i\leq k-1$. Let
\[
\bB_1=\{x\in \bB\setminus\{x_1\},\ x_1\in[x_0,x]\ \textrm{and}\ x_1\in[x_0,\varphi(x)]\}.
\]
Then there exists an extension $\cS'=\langle \bA',\varphi'\colon \bB'\to\bC'\rangle$ of $\cS$ such that $\bA'=\bA\cup\{y\}$, $\bB'=\bB\cup\{y\}$, $\bC'=\bC\cup\{y\}$ and $\varphi'(y)=y$. Moreover, for any $b\in \bB_1, b'\in \bB\setminus \bB_1$, $y\in [b,b']$.
\end{lem}
\begin{proof} Let us identify $\bA$ with a subset of $\Br(D_\infty)$ and let $g\in G_\infty$ such that $\varphi$ is the restriction of $g$ on $\bB$. By construction, $x_1$ belongs to the interior of an austro-boreal arc $I$ of $g$. This arc is contained in the union of two connected components of $D_\infty\setminus\{x_1\}$. Let $U$ be the one that does not contain $x_0$. By definition of $\bB_1$, $\bB_1\subset U$. Let choose a branch point $y$ in the interior of $I$ and in $U$ such that for any $b\in \bB_1$, $y\in [x_1,b]$ and $gy\neq x_1\in [x_1,y]$. By construction, no element of $\bB$ lies in $D(y,g(y))$. Choose a slightly larger arc $[z,z']$ in the interior of $I$ containing $[gy,y]$ and such that the preimage in $\bB\cup\bC$ of $[z,z']$ by the first point map to $I$ is empty. Let $h$ be a homeomorphism of $D(z',z)$ such that $h(g(y))=y$ and fixing $z,z'$. Let us extend $h$ to an element of $G_\infty$ by setting $h$ to be trivial outside $D(z,z')$. Now, let $g'=h\circ g$ and~$\varphi'$ to be its restriction on $\bB'=\bB\cup\{y\}$.
\end{proof}
\begin{prop}\label{prop:cofinal}The class $\cL$ is cofinal in $\cK_p$.
\end{prop}
\begin{proof}Let $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle\in\cK_p$. Thanks to Lemma~\ref{fixpt}, we may assume that $\varphi$ has a fixed point $x_0$.
Moreover, if $g\in G_\infty$ induces $\varphi$, one can replace $\bB$ by the $c$-closure of $\bB\cup \left(\bA\setminus \bB\cup\bC\right)$, $\bC$ by the image of this new $\bB$ by $g$ and $\bA$ the $c$-closure of $\bB\cup \bC$. Thanks to the patchwork lemma, we may reduce to the case where $\varphi$ has a unique orbit of branches $E$ around $x_0$ and thus $\cS=\cS_E$. Actually, we can deal with each orbit of branches separately and patch them together at the end. So let us assume that $\cS=\cS_E$ and let $D$ be given by Lemma~\ref{phi-orbit}.
The proof will be done thanks to different reductions and inductions.
\subsection*{Case A} The orbit of branches $E$ is reduced to $D$, that is, $n=1$ in Lemma~\ref{phi-orbit}. On $\bA\setminus\{x_0\}$, we define a partial order $x\leq y \ssi x\in[x_0,y]$. This is a semi-linear order (see \cite[\S 5]{DM_dendritesII}) and thus, since $\bB'=\bB\setminus\{x_0\}$ and $\bC'=\bC\setminus\{x_0\}$ are $c$-closed, they have a unique minimum that we denote respectively by $b_0$ and $c_0$. Since $\varphi$ preserves betweenness, we know that $\varphi(b_0)=c_0$. We set $\bF=\bB\cup\bC\cup\{c(x_0,b_0,c_0)\}$.
\subsubsection*{Subcase \textup{A.1:} $c(x_0,b_0,c_0)\notin\{b_0,c_0\}$} In this case, no point of $\bB'$ is between points of $\bC'$ and vice versa. In particular, they are disjoint. We can define an extension $\langle \bA,\psi\colon \bF\to\bF\rangle$, where $\psi|_\bB=\varphi$, $\psi(c(x_0,b_0,c_0))=c(x_0,b_0,c_0)$ and for $c\in\bC$, $\psi(c)=b\in\bB$ is the unique point $b\in\bB$ such that $\varphi(b)=c$. This extension belongs to $\cK_p$ and it satisfies Definition~\ref{class} with $\bF_0=\bB\cup\{c(x_0,b_0,c_0)\}$. Actually, for any $b\in\bF_0$, $\psi^2(b)=b$ and thus Condition \eqref{class2} is satisfied. Conditions \eqref{class3} \& \eqref{class4} are empty and $\bA$ is still the $c$-closure of $\bF$.
\subsubsection*{Subcase \textup{A.2:} $c(x_0,b_0,c_0)=c_0$}
Observe that for any $g\in G_\infty$ that induces $\varphi$, $b_0$ belongs to an austro-boreal arc for $g$ because $[b_0,x_0]$ is mapped to $[c_0,x_0]$ and thus $c_0=g(b_0)$ belongs to $[b_0,g^2(b_0)]$. Let us denote by $x_1,\dots, x_k$ the $\varphi$-orbit of $b_0$ such that $\varphi(x_i)=x_{i+1}$ (in particular $b_0=x_{k-1}$ and $c_0=x_k$). Let
\[
\bB_1=\{x\in \bB,\ x_1\notin{}]x,\varphi(x)[\ \textrm{and}\ x_1\in{}]x,x_0[\}.
\]
Thanks to Lemma~\ref{lem:addfixedpoint}, we may assume that $\varphi$ has a fixed point $y\in\bB_1$ between $x_1$ and any other point of $\bB\setminus\bB_1$. Let $\bC_1$ to be $\varphi(\bB_1)$ and $\bA_1$ to be $\{y\}$ and the union of the branches in $\bA$ around $y$ that do not contain $b_0$. In particular, $\bA_1$ is $c$-closed, contains $\bB_1\cup\bC_1$. We define $\cS_1=\langle\bA_1, \varphi_1\colon \bB_1\to\bC_1\rangle$, where $\varphi_1$ is the restriction of~$\varphi$ to $\bB_1$. By an induction on the number of $\varphi_1$-orbits which is less than the number of $\varphi$-orbits, we may assume that $\cS_1$ embeds in $\cS_1'\in\cL$.
Let us define $\bA_2=(\bA\setminus \bA_1)\cup\{y\}$, $\bB_2=\bB\cap \bA_2$, $\bC_2=\bC\cap\bA_2$ and let $\varphi_2$ be the restriction of $\varphi$ on $\bB_2$. Let choose $g\in G_\infty$ that induces $\varphi$. We set $\bB_2'$ to be~$\bB_2$ and we add successively points to this set. For any point $x$ in $\bB_2\cap{}]y,x_0[$, we know that $x\geq b_0$ and $g(x)\leq x_1$. We add to $\bB_2'$ all points $g^m(x),g^{m+1}(x),\dots,g^l(x)$, where $m$ is the maximal integer such that $g^m(x)\geq x_1$ and $l$ is the minimal integer such that $g^l(x)\geq b_0$. For $z\in \bB_2$ not in $[y,x_0]$, let $x_z\in{}]y,x_0[$ be $c(z,x_0,y)\in \bB_2$. Let $m,l$ be the corresponding integers for $x_z$. We add $\{g^m{z},\dots,g^l{z}\}$ to $\bB_2'$ in order to guarantee Condition \eqref{class5}. Finally, we replace $\bB'_2$ by its $c$-closure (this adds only finitely many points). Let $\bC_2'=g(\bB_2')$. Let us define $(\bB_2')_0=\bigl(\bB_2'\cap \overline{D(y,x_1)}\bigr)\cup\{x_0\}$. Up to adding $g^i((\bB_2')_0)$ to $\bB'_2$ for $i=1,2,3,4$ (and $g^i((\bB_2')_0)$ to $\bC'_2$ for $i=2,3,4,5$) we may assume that $l-m>4$. Now, let $\bA_2'$ be the $c$-closure of $\bA_2\cup\bB_2'\cup\bC_2'$ and $\varphi'_2$ be $g|_{\bB_2'}$. The system $\langle \bA_2, \varphi_2\colon\bB_2\to\bC_2\rangle$ embeds in $\cS_2'=\langle \bA_2', \varphi_2'\colon\bB_2'\to\bC_2'\rangle\in\cK_p$. Moreover, the latter one satisfies Condition \eqref{class2} in Definition \ref{class} with $n=1$ for all points. Actually, Condition \eqref{class1} is obtained by construction of $(\bB_2')_0$, $x_0$ and $y$ are fixed points and all the other points of $(\bB_2')_0$ are in $D(g)$ thus Conditions \eqref{class2} \& \eqref{class3} follow. For Condition \eqref{class4}, any two points of $\bB'$ that lie in $]y,x_0[$ have intertwined $\varphi$-orbit and the last possibility of Condition \eqref{class4} occurs. So $\cS'_2\in\cL$.
By the patchwork lemma, there exists $g\in D_\infty$ inducing $\varphi_1'$ and $\varphi_2'$. Thus, if~$\varphi'$ is the restriction of $g$ on $\bB_1\cup\bB_2$ $\cS'=\langle \bA_1\cup\bA_2',\ \varphi'\colon \bB_1\cup\bB_2'\to\bC_1\cup\bC_2'\rangle\in\cL$ is an extension of $\cS$.
\subsubsection*{Subcase \textup{A.3:} $c(x_0,b_0,c_0)=b_0$}
This subcase is very similar to Subcase A.2 and we only indicate what should be modified. The points $x_1,\dots x_k$ are the $\varphi$-orbit of $b_0$ but this time $x_1=b_0$ and $x_{i+1}=\varphi(x_i)$. Let
\[
\bB_1=\{x\in \bB,\ x_k\notin{}]x,\varphi(x)[ \ \textrm{and}\ x_k\in{}]x_0,x[\}.
\]
Thanks to Lemma~\ref{lem:addfixedpoint} applied to $\langle \bA,\varphi^{-1}\colon \bC\to\bB\rangle$, we may assume that $\varphi$ has a fixed point $y\in\bB_1$ between $x_k$ and any other point of $\bB\setminus\bB_1$. Let $\bC_1$ to be $\varphi(\bB_1)$ and~$\bA_1$ to be $\{y\}$ and the union of the branches in $\bA$ around $y$ that do not contain $b_0$. In~particular, $\bA_1$ is $c$-closed and contains $\bB_1\cup\bC_1$. We define \hbox{$\cS_1=\langle\bA_1, \varphi_1\colon \bB_1\to\bC_1\rangle$}, where $\varphi_1$ is the restriction of $\varphi$ to $\bB_1$. By an induction on the number of $\varphi_1$-orbits, we may assume that $\cS_1$ embeds in some $\cS_1'\in\cL$, where $\varphi_1$ is the restriction of $\varphi$ on~$\bB_1$.
Let us define $\bA_2=\bA\setminus \bA_1\cup\{y\}$, $\bB_2=\bB\cap \bA_2$, $\bC_2=\bC\cap\bA_2$ and let $\varphi_2$ be the restriction of $\varphi$ on $\bB_2$. Let choose $g\in G_\infty$ that induces $\varphi$. We set $\bB_2'$ to be $\bB_2$ and we add successively points to this set. For any point $x$ in $\bB_2\cap{}]y,x_0[$, we know that $x\geq b_0$ and $x\leq x_k$. We add to $\bB_2'$ all points $g^m(x),g^{m+1}(x),\dots,g^l(x)$ such that $l$ is the minimal integer such that $g^m(x)\geq x_k$ and $l$ is the minimal integer such that $g^l(x)\geq b_0$. For $z\in \bB_2$ not in $[y,x_0]$, let $x_z\in{}]y,x_0[$ be $c(z,x_0,y)\in \bB_2$. Let $m,l$ be the corresponding integers for $x_z$, we add $\{g^m{z},\dots,g^l{z}\}$ to $\bB_2'$. Finally, we replace~$\bB'_2$ by its $c$-closure (this adds only finitely many points). Let $\bC_2'=g(\bB_2')$. Let us define $(\bB_2')_0=\bigl(\bB_2'\cap \overline{D(x_1,x_2)}\bigr)\cup\{x_0\}$. Up to adding $g^i((\bB_2')_0)$ to $\bB'_2$ for $i=1,2,3,4$ (and $g^i((\bB_2')_0)$ to $\bC'_2$ for $i=2,3,4,5$) we may assume that $l-m>4$. Now, let $\bA_2'$ be the $c$-closure of $\bA_2\cup\bB_2'\cup\bC_2'$ and $\varphi'_2$ be the $g|_{\bB_2'}$. The system $\langle \bA_2, \varphi_2\colon\bB_2\to\bC_2\rangle$ embeds in $\cS_2'=\langle \bA_2', \varphi_2'\colon\bB_2'\to\bC_2'\rangle\in\cK_p$ and $\cS_2'\in\cL$ for the same reasons as above. We conclude similarly as in Subcase A.2.
\subsection*{Case B} The integer $n$ is larger than 1. The idea is then to reduce to Case A by making a precise definition for $\psi=\varphi^n$ and apply Case A to $\psi$.
\subsubsection*{Subcase \textup{B.1:} $\varphi^{n-1}(D)\cap \bB=\emptyset$} This case is quite similar to Subcase A.2.
Let us identify $\bA$ with a subset of $\Br(D_\infty)$ and $\varphi$ with the restriction of some $g\in\Homeo(D_\infty)$. Let $\widetilde{D}$ be the connected component of $D_\infty\setminus\{x_0\}$ that contains the points of $D$. Let $\bB_0$ be the $c$-closure of $\widetilde{D}\cap\bigl(\bigcup_{0\leq k< n-1}g^{-k}(\bB)\bigr)\cup\{x_0\}$. This is a finite set. Let $\bC'=\bB'=\bigcup_{0\leq k< n-1}g^k(\bB_0)$. We define $\varphi'$ to coincide with $g$ on $\bigcup_{0\leq k< n-1} g^k(\bB_0)$ and $g^{-(n-1)}$ on $g^{n-1}(\bB_0)$. The class $\cS'=\langle \bA',\varphi\colon\bB'\to\bC'\rangle$ which is an extension of $\cS$ belongs to $\cL$ with this integer $n$ and all points of $\bB_0$ are $\varphi^n$-fixed points.
\subsubsection*{Subcase \textup{B.2:} $\varphi^n(D)\cap \bB\neq\emptyset$} So $\varphi^n(D)=D$. Up to choosing $g\in G_\infty$ and add points in each $\varphi$-orbits, we may assume that all $\varphi$-orbits start, finish in $D$ and have all the same length that is, at least $4n$. That is, we may assume that $\bB_0=D\cap\bB$ satisfies $\bigcup_{0\leq k \leq ln-1}\varphi^k(\bB_0)=\bB$ with $l\geq4$ and $\bB_0$ is $c$-closed. This will guarantee the very last point of Condition \eqref{class4} in Definition~\ref{class}. We define the system $\cT=\langle \bA\cap D\cup\{x_0\},\psi\colon\bigcup_{0\leq k \leq l-1}\varphi^{kn}(\bB_0)\to\bigcup_{0\leq k \leq l-1}\varphi^{(k+1)n}(\bB_0)\rangle$, where $\psi$ is the restriction of $\varphi^n$ to $\bigcup_{0\leq k \leq l-1}\varphi^{kn}(\bB_0)$. Observe that the number of $\psi$-orbits is at most the same number of $\varphi$-orbits. Now, $\cT\in \cK_p$ and falls in Case A. So we can find an embedding of $\cT$ in some $\cT'=\langle \bA',\psi'\colon \bB'\to\bC'\rangle\in\cL$. Let us define $\bB''=\bigcup_{0\leq i1$. Since $\cS\in\cL$, $g$ has some fixed point $p\in\bB$. Since $m>1$, $y$ is not in the element of $\cC_x$ that contains $p$. So, if $k$ is not a multiple of $n$, then $c(p,x,g^k(x))$ separates $D(x,y)$ and $g^k(D(x,y))$. Thus $D(x,y)$ and $g^k(D(x,y))$ are disjoint. Now, if $D(x,y)$ and $g^k(D(x,y))$ are not disjoint then the point $c(x,y,g^k(y))$ is necessarily a non-periodic point because otherwise it would belong to $\bB$ which is $c$-closed. By assumption there is no such point.
\end{proof}
\begin{lem}\label{lem:fix}Let $g\in G_\infty$, let $x,y$ be $g$-fixed points in $D_\infty$ and let $M$ be some finite set in $]x,y[$. There are $z\in{}]x,y[$ such that $M\subset]x,z[$, $q\in{}]z,y[$ and $g'\in G_\infty$ that is equal to $g$ on $D_\infty\setminus D(z,y)$ and that fixes $q$.
\end{lem}
\begin{proof}Since $M$ is finite, one can find $z\in{}]x,y[$ such that $M\subset]x,z[$. Now, choose $q\in{}]z,y[{}\cap{}]gz,y[{}\subset]x,y[$. Find a homeomorphism $f$ from $D(z,y)$ to $D(gz,y)$ fixing $q,y$ and such that $f(z)=g(z)$ (this is possible thanks to \cite[Prop.\,6.1]{DM_dendritesII}). Now, define, $g'$ to be $f$ on $D(z,y)$ and $g$ elsewhere.
\end{proof}
\begin{lem}\label{lem:per} Let $\cS=\langle \bA,\varphi\colon\bB\to\bC\rangle\in\cL$ and $\cS\to \cT=\langle \bD,\psi\colon \bE\to\bF\rangle$ be an embedding. Let $x,y\in\bB$ be $\varphi$-periodic points. Assume that $]x,y[{}\cap\bB$ does not contain any periodic point. Then there is an embedding $\cT\to \cT'=\langle \bD',\psi'\colon \bE'\to\bF'\rangle$ such that there is $q\in \bD'$ with $\bD'=\bD\cup\{q\}$, $\bE'=\bE\cup\{q\}$, $\bF'=\bF\cup\{q\}$, $q\in{}]x,y[$, $z$ is $\psi'$-periodic and $]q,y[{}\cap\bD=\emptyset$.
\end{lem}
\begin{proof} Let $h\in G_\infty$ that induces $\psi$ and let us consider $\bD$ as a subset of $\Br(D_\infty)$. As in the proof of Lemma \ref{period}, let $m$ be the maximal period of $x$ and $y$. So, the subsets $h^k(D(x,y))$ are disjoint for $0 \leq k=triangle 45,x=1.0cm,y=1.0cm]
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\draw [line width=1.pt] (1.,2.)-- (1.5,3.);
\draw [line width=1.pt] (1.,2.)-- (0.5,3.);
\draw [line width=1.pt] (2.,1.)-- (1.,2.);
\draw [line width=1.pt] (2.,1.)-- (3.,2.);
\draw [line width=1.pt] (3.,2.)-- (2.5,3.);
\draw [line width=1.pt] (3.,2.)-- (3.5,3.);
\begin{scriptsize}
\draw [fill=black] (2.,0.) circle (2pt);
\draw[color=black] (2.3,0) node {$x_0$};
\draw [fill=black] (2.,1.) circle (2pt);
\draw[color=black] (2,1.3) node {$c$};
\draw [fill=black] (1.,2.) circle (2pt);
\draw[color=black] (1.3,2) node {$a$};
\draw [fill=black] (1.5,3.) circle (2pt);
\draw [fill=black] (0.5,3.) circle (2pt);
\draw[color=black] (0.5,3.3) node {$b$};
\draw [fill=black] (3.,2.) circle (2pt);
\draw[color=black] (3.3,2) node {$a'$};
\draw [fill=black] (2.5,3.) circle (2pt);
\draw [fill=black] (3.5,3.) circle (2pt);
\draw[color=black] (3.5,3.3) node {$b'$};
\end{scriptsize}
\end{tikzpicture}
\caption{The simple dendrite $D$ with a converging but not convex linear order $\prec$.}\label{notconvex}
\end{figure}
\begin{itemize}
\item $x_0$ is the root,
\item $c\prec a,a',b,b'$,
\item $a,a'\prec b,b'$,
\item $a\prec a'$,
\item $b'\prec b$.
\end{itemize}
The conditions $a\prec a'$ and $b'\prec b$ show that this order is not convex.
\end{rem}
We denote by $\CLO(X)$ the set of convex converging linear orders on $\Br(X)$. It is clear from the definition that $\CLO(X)$ is a metrizable $\Homeo(X)$-flow since it is a closed invariant subspace of the space of all linear orders $\LO(\Br(X))$ on $\Br(X)$ which is compact for the pointwise convergence.
We observe that a convex converging linear order $\prec$ induces a linear order $\prec^x$ on the connected components around a given point $x$.
\begin{lem}\label{obsorder}Let $x\in X$, ${\prec}{}\in \CLO(X)$ with root $x_0$. Let $C,C'\in\cC_x$ distinct and that do not contain the root. Then,
\[
(\forall c\in C,\ \forall c'\in C',\ c\prec c')\lor (\forall c\in C,\ \forall c'\in C',\ c\succ c').
\]
\end{lem}
\begin{proof} Let $y_0$ be the image of the root $x_0$ by the first point map to the subdendrite $C\cup\{x\}\cup C'$. Since $x_0\notin C\cup C'$, $y_0=x$.
Choose $a\in C$ and $b\in C'$. Assume that $a\prec b$. Now, by convexity of the order, for any $k\in C$ and $k'\in C'$, let $a'=c(k,a,x)$ and $b'=c(b,k',x)$. By convexity, $a'\prec b'$ and thus $k\prec k'$.
\end{proof}
In the first case, we write $C\prec^x C'$ and otherwise we write $C'\prec^x C$. This defines a linear order on $\cC_x$ if $x=x_0$ and on $\cC_x\setminus C_x(x_0)$ if $x\neq x_0$.
\begin{rem}Let us observe that convex and converging linear orders have the following stability property: If ${\prec}\in\CLO(X)$ and $Y$ is a subdendrite of $X$ then ${\prec}|_{\Br(Y)}\in\CLO(Y)$. If $x_0$ is the root of $\prec$ then $\pi_Y(x_0)$ is the root of $\prec |_{\Br(Y)}$.
\end{rem}
We will see in Theorem~\ref{univminflow} that the universal minimal $G_\infty$-flow is $\CLO(D_\infty)$. For the remaining of this section, we fix some $\xi\in\Ends(D_\infty)$. For a branch point $c$, let us denote by $\cC_{c,\xi}$ the space $\cC_c\setminus C_c(\xi)$.
\begin{lem}\label{cclo} For each branch point $c$, fix a linear order $\prec^c$ on the set $\cC_{c,\xi}$ that is isomorphic to $\QQ$ with its standard linear order <. Then there is a convex converging linear order $\prec _0$ on $D_\infty$ such that the root is $\xi$ and for any branch point $c$, the linear order induced on the components of $\cC_{c,\xi}$ is $\prec^c$.
\end{lem}
\begin{proof} We define $\prec_0$ in the following way: for $a\neq b\in\Br(D_\infty)$, if $c(a,b,\xi)=a$ then $a\prec_0 b$ (and $b\prec_0 a$ if $c(a,b,\xi)=b$). If $c=c(a,b,\xi)\neq a,b$ then $a\prec_0b\ssi C_c(a)\prec^c C_c(b)$.
Let us check it is a convex converging linear order. Totality and antisymmetry are immediate. Let $a,b,c\in\Br(D_\infty)$ such that $a\prec_0 b\prec_0 c$ and let us note $d=c(a,b,\xi)$ and $e=c(b,c,\xi)$. There are three (mutually exclusive) possibilities: $e\in{}]\xi,d[$, $d=e$ or $e\in{}]d,b]$. In the first case $a\in C_e(b)$, $C_e(a)\prec^e C_e(c)$. In the second one, $a\in[\xi,c[$ (if $a=d$) or $C_d(a)\prec^d C_d(e)$ and in the last one $C_d(a)\prec^d C_d(c)$. So, in all cases, $a\prec_0 c$.
This order is converging because if $a,b\in\Br(D_\infty)$ and $c=c(a,b,\xi)$ then $a$ is maximum on $[c,a]$ and $b$ is a maximum on $[c,b]$. Finally, this order is convex by construction.
\end{proof}
\begin{rem}The order $\prec_0$ depends a priori on the choice of the linear orders $\prec^c$ for all branch points. Actually, a different choice of orders isomorphic to $(\QQ,<)$ leads to an order $\prec$ such that there exists $g\in G_\infty$ fixing $\xi$ with $x\prec y\ssi g(x)\prec_0 g(y)$ for all $x,y\in\Br(D_\infty)$. This can be obtained thanks to a back and forth argument on $\Br(D_\infty)$. In what follows, we will not use that fact and we will fix the order $(\prec^c)_{c\in\Br(D_\infty)}$ in the proof of Proposition~\ref{fraise} and thus we will forget the dependency on this choice.\end{rem}
\begin{prop}\label{Ramsey} The group $\Stab_{G_\infty}(\prec _0)$ is extremely amenable.
\end{prop}
To prove the extreme amenability of this group, we used the seminal idea that a closed subgroup of $S_\infty$ is extremely amenable if and only if it is the automorphism group of some Fraïssé limit of a Fraïssé order class with the Ramsey property \cite[Th.\,4.7]{MR2140630}. We now describe the Fraïssé class and the Ramsey theorem needed to prove Proposition~\ref{Ramsey}. We essentially follow \cite{MR3436366}.
A (meet) \emph{semi-lattice} is a poset $(A,\leq)$ such that for any two elements $a,b\in A$, the pair $\{a,b\}$ has a greatest lower bound (that is, an infimum) denoted by $a\wedge b$ and called the \emph{meet} of $a$ and $b$. It satisfies the following three properties for all $a,b,c\in A$:
\begin{itemize}
\item $a\wedge a=a$,
\item $a\wedge b=b\wedge a$ and
\item $(a\wedge b)\wedge c=a\wedge(b\wedge c)$.
\end{itemize}
Actually, from a binary operation $\wedge$ satisfying the above three properties, one can recover the partial order $\leq$ by defining $a\leq b\ssi a\wedge b=a$.
A semi-lattice $(A,\leq,\wedge)$ is \emph{treeable} if it has a minimum called the \emph{root} and all the sets $\da a=\{b\in A;\ b\leq a\}$ are linearly ordered.
A linear order $\prec$ on a treeable semi-lattice $(A,\leq,\wedge)$ is a \emph{linear extension} of $\leq$ if $a< b\implique a\prec b$ and it is \emph{convex} if for any $a,a',b,b'\in A$ such that $a\wedge b\preceq a'\preceq a$ and $a\wedge b\preceq b'\prec b$, $a\prec b\ssi a'\prec b'$.
We denote by $\cCT$ the class of finite treeable semi-lattices with a convex linear extension $(A,\leq,\wedge,\prec)$.
\begin{rem} If $(A,\leq_A,\wedge_A,\prec_A)$ and $(B,\leq_B,\wedge_B,\prec_B)$ are elements of $\cCT$, an embedding of $A$ in $B$ is an injective map $\varphi\colon A\to B$ such that for all $a,a'\in A$, $\varphi(a\wedge_A a')=\varphi(a)\wedge_B\varphi(a')$ and $a\prec_A a'\implique \varphi(a)\prec_B\varphi(a')$.
We emphasize that this notion of embeddings does not coincide with the notion of embeddings for graphs. In our situation, one can add a vertex in the middle of an edge and this is impossible for graphs embeddings.
\end{rem}
Let us introduce the following partial order on $\Br(D_\infty)$: $a\leq b\ssi a\in[\xi,b]$.
\begin{lem}\label{lem:tr} The poset $(\Br(D_\infty),\leq)$ is a treeable semi-lattice and $\prec_0$ is a convex linear extension of $\leq$.
\end{lem}
\begin{proof} It is straightforward to check that it is a treeable semi-lattice with meet $a\wedge b=c(a,b,\xi)$ for any $a,b\in\Br(D_\infty)$. The fact that $\prec_0$ is a convex linear extension of $<$ follows from the properties given in Lemma~\ref{cclo}.
\end{proof}
\begin{prop}\label{fraise}The Fraïssé limit of $\cCT$ is $(\Br(D_\infty),\leq,\wedge,\prec_0)$.\end{prop}
To prove this proposition, we rely on the relation between semi-linear orders and dendrite with a chosen end point developed in \cite[\S 5]{DM_dendritesII}. A partially ordered set $(X,\leq)$ is a \emph{semi-linear} order if for any $x,y\in X$, there exists $z\in X$ such that $z\leq x,y$ and for all
$x\in X$, the downward chain $\da x=\{y\in X,\ y\leq x\}$ is totally ordered. Treeable semi-lattices are particular cases of semi-linear orders.
A partially ordered set $(X,\leq)$ is \emph{dense} if for all $x,y$ such that $xc$, are in the same connected of $\cC_c$ if and only if $c(\xi,a,b)\neq c$ that is, if and only if $a\wedge b>c$. Since $\prec$ is convex, it induces a dense linear order $\prec^c$ on elements of $\cC_c$ that do not contain $\xi$. By~the amalgamation property each order $\prec^c$ is countable and dense thus isomorphic to $(\QQ,<)$ and $\prec_0$ is obtained by Lemma~\ref{cclo}.
\end{proof}
\begin{lem}\label{groupiso} The groups $\Aut(\Br(D_\infty),\leq,\wedge,\prec_0)$ and $\Stab_{G_\infty}(\prec_0)$ are isomorphic.\end{lem}
\begin{proof} It is proved in \cite[Cor.\,5.21]{DM_dendritesII} that $\Aut(\Br(D_\infty),\leq)\simeq \Stab_{G_\infty}(\prec_0)$ and the lemma follows.
\end{proof}
We can now prove Proposition~\ref{Ramsey}.
\begin{proof} [Proof of Proposition~\ref{Ramsey}] Thanks to \cite[Th.\,4.7]{MR2140630}, it suffices to show that the group $\Stab_{G_\infty}(\prec_0)$ is the automorphism group of some Fraïssé limit of some Fraïssé order class with the Ramsey property. By Lemma~\ref{groupiso}, $\Stab_{G_\infty}(\prec_0)$ is the automorphism group of the limit of the class $\cCT$ and this class has the Ramsey property \cite[Th.\,2]{MR3436366}.
\end{proof}
Let us denote by $\CLO(D_\infty)_{\xi}$ the closed subspace of $\CLO(D_\infty)$ of convex converging linear orders with root $\xi$. For brevity we denote $G_{\xi}=\Stab_{G_\infty}(\xi).$
\begin{lem}\label{minimality}Any $G_\infty$-orbit in $\CLO(D_\infty)$ is dense. Similarly, any $G_{\xi}$-orbit in $\CLO(D_\infty)_{\xi}$ is dense.
\end{lem}
\begin{proof}One has to show that for any pair $\prec_1,{\prec_2}\in\CLO(D_\infty)$ and any finite subset $F\subset\Br(D_\infty)$, there is $g\in G_\infty$ that induces an isomorphism from $(F,\prec_1)$ to $(gF,\prec_2)$ (\ie for any $x,y\in F$, $x\prec_1 y\ssi g(x)\prec_2g(y)$); and moreover if $\xi$ is the root of $\prec_1$ and $\prec_2$ then $g$ can be chosen in $G_{\xi}$.
For any finite set $F$ in a dendrite, the subdendrite $[F]$, that is, the smallest subdendrite containing $F$, has finitely many branch points. So, adding these branch points to $F$ if required, we assume that $F$ is $c$-closed. We proceed by induction on the cardinality of $F$. If $F$ is reduced to a point then the result is immediate because $G_{\xi}$ acts transitively on branch points. Assume $F$ has $n\geq2$ points and we have the result for~$n-1$. Let $m$ be the maximum of $F$ for $\prec_1$. The converging property of $\prec_1$ implies that $m$ is an end point of $[F]$ and thus $F'=F\setminus\{m\}$ is also a finite $c$-closed subset of $\Br(D_\infty)$ and thus by induction there exists $g_1\in G_\infty$ that induces an isomorphism from $(F',\prec_1)$ to $(g_1(F'),\prec_2)$. Moreover, if $\prec_1,\prec_2$ have root $\xi$, then $g_1\in G_{\xi}$. It remains to put $m$ in the right position.
\begin{claim*}
Let $\prec$ be some convex converging linear order on $\Br(D_\infty)$, $x_1,x_2\in\Br(D_\infty)$ and $C_i\in\cC_{x_i}$ such that $C_i$ does not contain the root of $\prec$. If $F$ is a finite $c$-closed subset such that $F\subset \overline{C_1}$ and $x_1\in F$ then there exists a homeomorphism $h$ from~$\overline{C_1}$ to~$\overline{C_2}$ such that $h(x_1)=x_2$ and $h$ is increasing for $\prec$ on $F$.
\end{claim*}
\begin{proof}[Proof of the claim]
Once again, we argue by induction and the case where $F=\{x_1\}$ is simply the fact there exists a homeomorphism from $\overline{C_1}$ to $\overline{C_2}$ that maps~$x_1$ to~$x_2$. So, assume $F$ has cardinality at least 2. Since $F'=F\setminus\{x_1\}$ is included in~$C_1$, there exists a minimal point $x'_1\in F'$ such that for any $y\in F$, $x'_1\in[x_1,y]$ and this point is in fact the minimum of $F\setminus\{x_1\}$. Let $C^1_1,\dots,C^k_1$ be the connected components of $C_1\setminus\{x_1'\}$ that meet $F'$ and let us denote $F_i=C^i_1\cap F'\cup\{x'_1\}$. We assume that these components are numbered increasingly ($C^i_1\prec^{x'}C^j_1\ssi i