Journal de l'École polytechnique
Mathématiques

Bruno DUCHESNE

Erratum to “Topological properties of Ważewski dendrite groups”

Tome II (2024), p. 1029-1034.

https://doi.org/10.5802/jep.272

© L'auteur, 2024.

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ERRATUM TO “TOPOLOGICAL PROPERTIES OF
WAŻEWSKI DENDRITE GROUPS”

BY BRUNO DUCHESNE

ABSTRACT. — In [Duc20], the universal minimal flow of the group of homeomorphisms of the universal Ważewski dendrite was identified as the completion of some coset space. A more concrete description was erroneously given in that paper as Basso and Tsankov explained in [BT23]. The aim of this note is to give a corrected description of this universal minimal flow.

More Precisely, Theorem 1.12 in [Duc20] (that is Theorem 7.16 with a more precise statement in the body of the article) is wrong and should be replaced by Theorem 2 and Theorem 4 below. The mistake is located in the proof of Theorem 7.16 and the other statements are not affected by the mistake.

RÉSUMÉ (Erratum à « Propriétés topologiques des groupes d'homéomorphismes des dendrites de Ważewski »)

Dans [Duc20], le flot minimal universel du groupe des homéomorphismes de la dendrite universelle de Ważewski a été identifié comme la complétion d'un certain espace de classes d'équivalence. Une description plus concrète a été donnée par erreur dans cet article, comme Basso et Tsankov l'ont expliqué dans [BT23]. Le but de cette note est de donner une description corrigée de ce flot minimal universel.

Plus précisément, le théorème 1.12 dans [Duc20] (qui est le théorème 7.16 avec un énoncé plus précis dans le corps de l'article) est erroné et devrait être remplacé par le théorème 2 et le théorème 4 ci-dessous. L'erreur se situe dans la preuve du théorème 7.16 et les autres énoncés ne sont pas affectés par l'erreur.

CONTENTS

<table><tr><td>1. Corrected statement</td><td>1029</td></tr><tr><td>2. Proofs</td><td>1031</td></tr><tr><td>References</td><td>1034</td></tr></table>

1. CORRECTED STATEMENT

We use the same notations as in [Duc20] and refer to it for the context. So $D_\infty$ is the universal Ważewski dendrite, that is the unique dendrite with dense branch points set and such that any branch point has infinite order. Let $G$ be its homeomorphism

MATHEMATICAL SUBJECT CLASSIFICATION (2020). — 22F50, 57S05, 37B05.

KEYWORDS. — Ważewski dendrites, groups of homeomorphisms, Polish groups, Steinhaus property, generic elements, automatic continuity, universal flows.
group. We denote by $\text{CCLO}(D_\infty)$ the set of convex and converging linear orders on the set of branch points $\text{Br}(D_\infty)$ [Duc20, Def. 7.3]. Basso and Tsankov gave a simpler definition [BT23, §9] of its elements. Let us recall it:

**DEFINITION 1.** — Let $\prec$ be a linear order on the set $\text{Br}(D_\infty)$.

— The order $\prec$ is converging if for any three distinct branch points $x_1, x_2, x_3$ such that $x_1 \prec x_3 \prec x_2$ then $x_2$ is not between $x_1$ and $x_3$.

— A converging order is, moreover, convex if for any points $x_1, x_2, x_3, x_4$ such that $x_2 \in (x_1, x_3)$ and $x_3 \in (x_2, x_4)$ then it is not the case that $x_2 \prec x_3 \prec x_1 \prec x_4$.

Let us recall that for a converging linear order on a dendrite, any minimizing sequence has a limit and this limit does not depend on the minimizing sequence [Duc20, Lem. 7.2]. It is called the root of the linear order.

The space $\text{CCLO}(D_\infty)$ of convex converging linear orders is a closed $G$-invariant subspace of the space of all linear orders on all branch points and thus a $G$-flow. In [Duc20, Th. 1.12], it is stated that $\text{CCLO}(D_\infty)$ is the universal minimal flow of $G$. This is false as Basso and Tsankov proved in [BT23, Th. 9.3]. Among convex converging linear orders, there is a generic $G$-orbit. The elements of the generic orbit can be described in the following way: the root is some end point $\xi$ and for each branch point $b$, the linear order induced on branches around $b$ not containing $\xi$ is isomorphic to the standard order on rational numbers. Let us fix one of these convex converging linear orders and denote it $\prec_0$. Its stabilizer is denoted $G_{\prec_0}$.

Let $K$ be the blown-up dendrite $\overline{\text{Ends}(D_\infty)}$, that is the completion of the space of ends of $D_\infty$ with respect to right uniformity when $\text{Ends}(D_\infty)$ is identified with $G/G_\xi$ where $G_\xi$ is the stabilizer of an end point $\xi \in D_\infty$ in $G$. This appears to be the universal Furstenberg boundary of $G$ [Duc20, §8.2].

An element $C$ of $K$ can be identified with a collection $(C_b)_{b \in \text{Br}(D_\infty)}$ where $C_b$ is an element of $\bar{b} = \hat{b} \cup \{\infty\}$, the one-point compactification of the discrete space $\hat{b}$ of branches around $b \in \text{Br}(D_\infty)$.

We introduce the following subset of $\text{CCLO}(D_\infty) \times K$:

$$M = \{(\prec, C) \in \text{CCLO}(D_\infty) \times K, \forall b, b' \in \text{Br}(D_\infty), b \prec b' \implies b \in C_b'\}.$$

Since $\text{CCLO}(D_\infty)$ and $K$ are $G$-flows, it follows that $M$ is a $G$-flow as well since it is invariant and closed.

For a uniform space $X$, we denote by $\hat{X}$ its completion. For example, $K = \overline{G/G_\xi}$. We can now give the correct concrete identification of the universal minimal flow of $G$.

**THEOREM 2.** — The universal minimal flow of $G$ is $M \simeq \overline{G/G_{\prec_0}}$.

**REMARK 3.** — This universal minimal flow can also be recovered using highly proximal extensions. Since $G \curvearrowleft \text{CCLO}(D_\infty)$ is minimal and highly proximal extensions preserve minimality [AG77], $G \curvearrowleft S_G(\text{CCLO}(D_\infty))$ (the universal highly proximal extension of $\text{CCLO}(D_\infty)$) is minimal. By [BT23, Prop. 2.7], $S_G(\text{CCLO}(D_\infty)) \simeq \text{Sa}(G/G_{\prec_0})$
(the Samuel compactification of $G/G_{<0}$), so by [Zuc21, Prop. 6.6], $G_{<0}$ is pre-syndetic. Since $G_{<0}$ is also extremely amenable [Duc20, Prop. 7.9], by [Zuc21, Th. 7.5],

$$M(G) \simeq \mathrm{Sa}(G/G_{<0}) \simeq S_G(\mathrm{CCLO}(D_\infty)).$$

The mistake in the proof of [Duc20, Th. 7.16] is located in the statement that the embedding $G/G_\xi \simeq \mathrm{CCLO}(D_\infty)$ is bi-uniformly continuous. It is uniformly continuous but its inverse is not. Let us conclude with points in [Duc20] that depend on this theorem and see how they are affected.

Corollary 7.17 in [Duc20] still holds since $M$ is metrizable (and thus has a comeager orbit). The universal minimal flow of $G_\xi$ is as depicted in [Duc20]. Let us denote by $M_\xi = \{m \in M, \pi_1(m) = \xi\}$ where $\pi_1(<, C) \in D_\infty$ is the root of $<$.

**THEOREM 4.** — The universal minimal flow of $G_\xi$ is indeed $M_\xi \simeq \mathrm{CCLO}(D_\infty)_\xi$.

This result is used in the proof of the amenability of $G_\xi$ [Duc20, Th. 8.5] and this amenability result is unaffected.

*Acknowledgements.* — I warmly thank Gianluca Basso and Todor Tsankov for explanations about the mistake and more specifically Gianluca Basso for discussions that follow.

## 2. PROOFS

### Proof of Theorem 2

*Identification between $M$ and $\widehat{G/G_{<0}}$. —* For $\xi \in \mathrm{Ends}(D_\infty)$, we denote by $C^\xi$ the element of $K$ such that for any $b \in \mathrm{Br}(D_\infty)$, $C_b^\xi$ is the branch around $b$ containing $\xi$ (denoted $C_b(\xi)$ in the original paper). Let $\xi_0$ be the root of $<$ and $C^0 = C^{\xi_0}$. Let us consider the orbit map $g \mapsto (g <_0, gC^0)$ from $G$ to $M$. On the image in $M$ of this orbit, a basis of entourages for the uniform structure is given by entourages

$$V_F = \{((<_1, C^{\xi_1})), (<_2, C^{\xi_2})) : \forall x \neq y \in F, x <_1 y \Leftrightarrow x <_2 y, \exists \gamma \in U_F, \gamma\xi_1 = \xi_2\},$$

where $F$ is some finite subset of $\mathrm{Br}(D_\infty)$ and $U_F$ is its pointwise stabilizer. Moreover, it suffices to consider finite $c$-closed subsets $F$ since the family given by these subsets is cofinal for inclusion. The description of the uniform structure comes from the identification proved in [Duc20, Prop. 8.13]. Since $G_{<0}$ fixes $\xi_0$ and thus $C^0$, this orbit map induces a map $G/G_{<0} \to M$ that is uniformly continuous and thus extends to an equivariant continuous map $\widehat{G/G_{<0}} \to M$. Let us see that the inverse map from the $G$-orbit of $(<_0, C^0)$ to $G/G_{<0}$ is uniformly continuous as well.

The left uniform structure on the quotient space $G/G_{<0}$ is generated by the system of basic entourages $E_F = \{(gG_{<0}, hG_{<0}), g \in U_F, hG_{<0} \} \text{ where } F \text{ is a finite subset of } \mathrm{Br}(D_\infty) \text{ as above.}$

So let us fix some finite subset $F \subset \mathrm{Br}(D_\infty)$ and let us prove that for two elements $g, h \in G$, if $((g <_0, g\xi_0), (h <_0, h\xi_0)) \in V_F$ then $(gG_{<0}, hG_{<0}) \in E_F$. This will prove the uniform continuity of the inverse of the orbit map $G/G_{<0} \to M$ on its image.
In that case by the description of $V_F$, there is $u \in U_F$ such that $g\xi_0 = uh\xi_0$. Then, applying Lemma 5 (see below) to $\prec = uh \prec_0$ and $\prec' = g \prec_0$, there is $v \in G$ fixing pointwise $F$ and $\xi = g\xi_0$ such that $g \prec_0 = vuh \prec_0$ and thus for $u' = vu \in U_F$ one has $g \in u'hG_{\prec_0}$ as required.

*Minimality.* — Let $(\prec_1, C^1), (\prec_2, C^2) \in M$. We aim to show that the closure of the $G$-orbit of $(\prec_1, C^1)$ contains $(\prec_2, C^2)$. Fix $F \subset \text{Br}(D_\infty)$ finite and for each $b \in F$, choose a neighborhood $N_b$ of $C_b^2$. We may assume that $N_b$ is a singleton if $C_b^2 \neq \infty$. To prove that the closure of the $G$-orbit of $(\prec_1, C^1)$ contains $(\prec_2, C^2)$, it suffices to prove there is $g \in G$ such that $g \prec_1$ coincides with $\prec_2$ on $F$ and $gC_b^1 \in N_b$ for all $b \in F$.

Let $r_1$ be the root of $\prec_1$. Let $g \in G$ such that $gr_1 \in \bigcap_{b \in F} N_b$ (this intersection is non-empty since eventually it contains a minimizing sequence for $\prec_2$). Let $b_0$ be the projection (i.e., the image by the first-point map) of $gr_1$ on $[F]$. Let $g_1 \in G$ fixing pointwise the branch around $b_0$ containing $gr_1$ and permuting the branches around $b_0$ containing elements of $F$ in order that the orders induced on these branches coincide for $\prec_2$ and $g_1g \prec_1$. The closure $B_1, \dots, B_n$ of each of these branches is homeomorphic to $D_\infty$ itself and $b_0$ is an end point for these subdendrites. One can now use [Duc20, Lem. 7.14] for each $b_i$ and find an element $h_i \in \text{Homeo}(B_i)$ fixing $b_0$ such that $h_i g_1 g \prec_1$ and $\prec_2$ coincide on $F \cap B_i$. Patching the $h_i$'s (extended by the identity on the other branches around $b_0$), one gets an element $g_2$ such that $g_2 g_1 g \prec_1$ and $\prec_2$ coincides on $F$. Now, it suffices to observe that for each $b \in F$, $g_2 g_1 g C_b^1$ is the branch around $g_2 g_1 g b$ containing the root of $g_2 g_1 g \prec_1$ that is $C_b^2$ or lies in $N_b$ if $b$ is the root of $\prec_2$.

*Universality.* — Since $G_{\prec_0}$ is extremely amenable, for any $G$-flow $X$, there is a $G_{\prec_0}$-fixed point $x$. The orbit map $g \mapsto gx$ is uniformly continuous and thus extends to a $G$-map $\tilde{G}/\tilde{G}_{\prec_0} \to X$. If $X$ is moreover minimal, the image is $X$ itself. $\square$

**LEMMA 5.** — Let $\prec, \prec'$ be two convex converging linear orders with root $\xi \in \text{Ends}(D_\infty)$ in the $G$-orbit of $\prec_0$ and $F$ be a finite c-closed subset of $\text{Br}(D_\infty)$ such that for any $x, y \in F$, $x \prec y \Leftrightarrow x \prec' y$. Then there exists $v \in G_\xi$ fixing pointwise $F$ such that $v \prec = \prec'$.

*Proof.* — First, we may assume that $F$ contains the projection $p$ of $\xi$ to the subdendrite generated by $F$. Actually if this point $p$ does not belong to $F$, we observe that since these orders are converging and they have the same root, $p$ has to be smaller than any other element of $F$ for $\prec$ and $\prec'$.

We use a back and forth argument to construct $v$. Actually, we construct $v$ as a bijection of $\text{Br}(D_\infty)$ preserving the betweenness relation and use the identification of $G$ with the group of betweenness preserving bijections of $\text{Br}(D_\infty)$.

Let $\{x_n\}_{n \ge 1}$ be an enumeration of $\text{Br}(D_\infty) \setminus F$. We construct increasing sequences $X_n, Y_n$ of finite subsets of $\text{Br}(D_\infty)$ containing $F$ and bijections $v_n: X_n \to Y_n$ such that for any $n \in \mathbb{N}$:
(1) $X_n, Y_n$ are c-closed and contain the projection of $\xi$ on the subdendrite they generate.

(2) $x_n \in X_n \cap Y_n$,

(3) $v_n$ preserves the betweenness relation and for any $x, y \in X_n$,

$$x \prec y \iff v_n(x) \prec' v_n(y),$$

(4) if $p_n \in X_n$ is the projection of $\xi$ on the subdendrite generated by $X_n$ then $v_n(p_n)$ is the projection of $\xi$ on the subdendrite generated by $Y_n$,

(5) $v_{n+1}|_{X_n} = v_n$.

We set $X_0 = Y_0 = F$ and $v_0$ to be the identity on $F$. We proceed by induction. Assume for some $n \in \mathbb{N}$, $X_n, Y_n$ and $v_n$ have been defined and satisfy the above properties. Let $c_n$ be the projection of $x_{n+1}$ on the subdendrite generated by $X_n \cup \{\xi\}$. Let $[a_n, b_n]$ be the minimal arc (maybe reduced to $c_n$ if $c_n \in X_n$) such that $a_n, b_n \in X_n \cup \{\xi\}$ containing $c_n$. Let $c'_n$ be a branch point in $(v_n(a_n), v_n(b_n))$ (with the convention that $v_n(\xi) = \xi$) or $c'_n = v_n(a_n) = v_n(b_n)$ if $c_n \in X_n$.

Let us observe that for any $f \in X_n$, $c_n \prec f \iff c'_n \prec' v_n(f)$. Actually, let $d_n = c(\xi, c_n, f) \in X_n$, if $d_n = c_n$ then $c_n \prec d_n$, if $d_n = f$ then $f \prec c_n$ (by the converging property) and otherwise

$$f \prec c_n \iff f \prec a_n \iff f \prec b_n$$

(see [Duc20, Lem. 7.5]). These properties are true because $c_n \in [a_n, b_n]$ and thus, the same holds for any $c'_n \in [v_n(a_n), v_n(b_n)]$.

If $x_n = c_n$ then let $x'_n$ be $c'_n$ and otherwise choose a point $x'_n$ in some component not containing $\xi, v_n(a_n), v_n(b_n)$ around $c'_n$ such that for any $x \in X_n$,

$$x_{n+1} \prec x \iff x'_n \prec' v_n(x).$$

This is possible because $\prec'$ is in the $G$-orbit of $\prec_0$ and thus the linear order induced by $\prec'$ on branches around $c'_n$ not containing $\xi$ is isomorphic to the standard order on rational numbers. Let $Y'_n = Y_n \cup \{x'_n, c'_n\}$ and $X'_n = X_n \cup \{x_{n+1}, c_n\}$ and let us extend $v_n$ on $X'_n$ with $v_n(x_{n+1}) = x'_n$ and $v_n(c_n) = c'_n$.

Similarly let $d'_n$ be the projection of $x_{n+1}$ on the subdendrite generated by $Y'_n \cup \{\xi\}$. If $d'_n \notin Y'_n$, let $[a'_n, b'_n]$ be the minimal arc such that $a'_n, b'_n \in Y'_n \cup \{\xi\}$ and containing $d'_n$. Let $d_n$ be a branch point in $(v_n^{-1}(a'_n), v_n^{-1}(b'_n))$ and choose a point $y_n$ in some component around $d_n$ not intersecting $X'_n \cup \{\xi\}$ and such that for any $x \in X_n$,

$$y_n \prec x \iff x_{n+1} \prec' v_n(x).$$

Now let $X_{n+1} = X'_n \cup \{d_n, y_n\}$, $Y_{n+1} = Y'_n \cup \{d'_n, x_{n+1}\}$ and $v_{n+1}(x) = v_n(x)$ for all $x \in X'_n$, $v_{n+1}(d_n) = d'_n$ and $v_{n+1}(y_n) = x_{n+1}$ (if these new points do not already belong to $X'_n$).

By construction, $X_n, Y_n$ and $v_n$ have the announced properties and finally, for $x \in \text{Br}(D_\infty)$ we define $v(x) = v_n(x)$ for any $n$ large enough such that $x \in X_n$. The map $v$ is a bijection that preserves the betweenness relation and maps $\prec$ to $\prec'$. Because of the fourth property, $v$ fixes $\xi$. Thus $v$ fixes pointwise $F \cup \{\xi\}$. $\square$
*Proof of Theorem 4.* — Since $G$ acts transitively on $\text{Ends}(D_\infty)$, we continue with $\xi = \xi_0$ the root of $\prec_0$. The space $M_{\xi_0}$ is closed and thus compact since $\pi_1$ is continuous. Let $\pi$ be the projection $M \to \text{CCLO}(D_\infty)$. This map is continuous on $M$ and injective on $M_{\xi_0}$ (in that case, the second component is constant equal to $C^0$) with image $\text{CCLO}(D_\infty)_{\xi_0}$. Since $\text{CCLO}(D_\infty)_{\xi_0}$ and $M_{\xi_0}$ are both compact, these two spaces are homeomorphic as $G_{\xi_0}$-flows. The first part of the proof of Theorem 2 also proves that $G_{\xi_0}/G_{\prec_0}$ is bi-uniformly embedded in $M_{\xi_0}$ and since it is dense $G_{\xi_0}/\widetilde{G_{\prec_0}} \simeq M_{\xi_0}$. Minimality and universality go as in the proof of Theorem 2. $\square$

## REFERENCES

[AG77] J. AUSLANDER & S. GLASNER – “Distal and highly proximal extensions of minimal flows”, *Indiana Univ. Math. J.* **26** (1977), no. 4, p. 731–749.

[BT23] G. BASSO & T. TSANKOV – “Topological dynamics of kaleidoscopic groups”, *Adv. Math.* **416** (2023), article no. 108915 (49 pages).

[Duc20] B. DUCHESNE – “Topological properties of Ważewski dendrite groups”, *J. Éc. polytech. Math.* **7** (2020), p. 431–477.

[Zuc21] A. ZUCKER – “Maximally highly proximal flows”, *Ergodic Theory Dynam. Systems* **41** (2021), no. 7, p. 2220–2240.

Manuscript received 21st September 2023
accepted 3rd September 2024

BRUNO DUCHESNE, Université Paris-Saclay, CNRS, Laboratoire de mathématiques d'Orsay,
Bâtiment 307, 91405, Orsay, France

E-mail: bruno.duchesne@universite-paris-saclay.fr

Url: https://www.imo.universite-paris-saclay.fr/-bruno.duchesne/