Journal de l’École polytechnique. Mathématiques

Lorenzo Dello Schiavo

Ronan Herry

Kohei Suzuki

Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Mao Sheng

Hao Sun

Jianping Wang

On the existence of gr-semistable filtrations of orthogonal/symplectic $\lambda $-connections

Giovanni Bellettini

Roberta Marziani

Riccardo Scala

A non-parametric Plateau problem with partial free boundary

Gustave Billon

Moduli spaces of marked branched projective structures on surfaces

Marta Pieropan

Damaris Schindler

Hyperbola method on toric varieties

Antonin Monteil

Paul Pegon

Mass concentration in rescaled first order integral functionals

Vlerë Mehmeti

An analytic viewpoint on the Hasse principle

Bruno Duchesne

Bruno Duchesne
Erratum to “Topological properties of Ważewski dendrite groups”
Tome 11 (2024), p. 1029-1034.
https://doi.org/10.5802/jep.272
© L’auteur, 2024.
Cet article est mis à disposition selon les termes de la licence
LICENCE INTERNATIONALE D’ATTRIBUTION CREATIVE COMMONS BY 4.0.
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Publié avec le soutien
du Centre National de la Recherche Scientifique
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e-ISSN : 2270-518X
Tome 11, 2024, p. 1029–1034 DOI: 10.5802/jep.272
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
1. Corrected statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1029
2. Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1031
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1034
1. Corrected statement
We use the same notations as in [Duc20] and refer to it for the context. So D∞ 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.
e-ISSN: 2270-518X http://jep.centre-mersenne.org/
1030 B. Duchesne
group. We denote by CCLO(D∞) the set of convex and converging linear orders on
the set of branch points Br(D∞) [Duc20, Def. 7.3]. Basso and Tsankov gave a simpler
definition [BT23, §9] of its elements. Let us recall it:
Definition 1. — Let ≺ be a linear order on the set Br(D∞).
– The order ≺ is converging if for any three distinct branch points x1, x2, x3 such
that x1 ≺ x3 ≺ x2 then x2 is not between x1 and x3.
– A converging order is, moreover, convex if for any points x1, x2, x3, x4 such that
x2 ∈ (x1, x3) and x3 ∈ (x2, x4) then it is not the case that x2 ≺ x3 ≺ x1 ≺ x4.
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 CCLO(D∞) 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 CCLO(D∞) is the universal minimal flow of G.
This is false as Basso and Tsankov proved in [BT23, Th. 9.3]. Among convex converg-
ing 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 ξ and for each branch point
b, the linear order induced on branches around b not containing ξ is isomorphic to the
standard order on rational numbers. Let us fix one of these convex converging linear
orders and denote it ≺0. Its stabilizer is denoted G≺0 .
Let K be the blown-up dendrite Ends(D∞), that is the completion of the space
of ends of D∞ with respect to right uniformity when Ends(D∞) is identified with
G/Gξ where Gξ is the stabilizer of an end point ξ ∈ D∞ 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 (Cb)b∈Br(D∞) where Cb is
an element of b = b̂ ∪ {∞}, the one-point compactification of the discrete space b̂ of
branches around b ∈ Br(D∞).
We introduce the following subset of CCLO(D∞)×K:
M =
{
(≺, C) ∈ CCLO(D∞)×K, ∀b, b′ ∈ Br(D∞), b ≺ b′ =⇒ b ∈ C ′b
}
.
Since CCLO(D∞) 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 X̂ its completion. For example, K = Ĝ/Gξ.
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 ≃ Ĝ/G≺0 .
Remark 3. — This universal minimal flow can also be recovered using highly proximal
extensions. Since G ↷ CCLO(D∞) is minimal and highly proximal extensions pre-
serve minimality [AG77], G ↷ SG(CCLO(D∞)) (the universal highly proximal exten-
sion of CCLO(D∞)) is minimal. By [BT23, Prop. 2.7], SG(CCLO(D∞)) ≃ Sa(G/G≺0)
J.É.P. — M., 2024, tome 11
Erratum to “Topological properties of Ważewski dendrite groups” 1031
(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) ≃ Sa(G/G≺0) ≃ SG(CCLO(D∞)).
The mistake in the proof of [Duc20, Th. 7.16] is located in the statement that the
embedding G/Gξ ≃ CCLO(D∞) 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 Gξ is as depicted in [Duc20]. Let us denote by
Mξ = {m ∈ M, π1(m) = ξ} where π1(≺, C) ∈ D∞ is the root of ≺.
Theorem 4. — The universal minimal flow of Gξ is indeed Mξ ≃ CCLO(D∞)ξ.
This result is used in the proof of the amenability of Gξ [Duc20, Th. 8.5] and this
amenability result is unaffected.
Acknowledgements. — I warmly thank Gianluca Basso and Todor Tsankov for expla-
nations about the mistake and more specifically Gianluca Basso for discussions that
follow.
2. Proofs
Proof of Theorem 2
Identification between M and Ĝ/G≺0 . — For ξ ∈ Ends(D∞), we denote by Cξ the
element of K such that for any b ∈ Br(D∞), Cξb is the branch around b containing ξ
(denoted Cb(ξ) in the original paper). Let ξ0 be the root of ≺0 and C0 = Cξ0 . Let
us consider the orbit map g 7→ (g ≺0, gC0) from G to M . On the image in M of this
orbit, a basis of entourages for the uniform structure is given by entourages
VF = {((≺1, Cξ1), (≺2, Cξ2) : ∀x ̸= y ∈ F, x ≺1 y ⇔ x ≺2 y, ∃γ ∈ UF , γξ1 = ξ2},
where F is some finite subset of Br(D∞) and UF is its pointwise stabilizer. Moreover,
it suffices to consider finite c-closed subsets F since the family given by these sub-
sets is cofinal for inclusion. The description of the uniform structure comes from the
identification proved in [Duc20, Prop. 8.13]. Since G≺0 fixes ξ0 and thus C0, this orbit
map induces a map G/G≺0 → M that is uniformly continuous and thus extends to
an equivariant continuous map Ĝ/G≺0 → M . Let us see that the inverse map from
the G-orbit of (≺0, C0) 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 EF = {(gG≺0 , hG≺0), g ∈ UFhG≺0} where F is a finite subset of
Br(D∞) as above.
So let us fix some finite subset F ⊂ Br(D∞) and let us prove that for two elements
g, h ∈ G, if ((g ≺0, gξ0), (h ≺0, hξ0)) ∈ VF then (gG≺0 , hG≺0) ∈ EF . This will prove
the uniform continuity of the inverse of the orbit map G/G≺0 → M on its image.
J.É.P. — M., 2024, tome 11
1032 B. Duchesne
In that case by the description of VF , there is u ∈ UF such that gξ0 = uhξ0. Then,
applying Lemma 5 (see below) to ≺ = uh ≺0 and ≺′ = g ≺0, there is v ∈ G fixing
pointwise F and ξ = gξ0 such that g ≺0 = vuh ≺0 and thus for u′ = vu ∈ UF one
has g ∈ u′hG≺0 as required.
Minimality. — Let (≺1, C1), (≺2, C2) ∈ M . We aim to show that the closure of the
G-orbit of (≺1, C1) contains (≺2, C2). Fix F ⊂ Br(D∞) finite and for each b ∈ F ,
choose a neighborhood Nb of C2b . We may assume that Nb is a singleton if C2b ̸= ∞.
To prove that the closure of the G-orbit of (≺1, C1) contains (≺2, C2), it suffices to
prove there is g ∈ G such that g ≺1 coincides with ≺2 on F and gC1b ∈ Nb for all
b ∈ F .
Let r1 be the root of ≺1. Let g ∈ G such that gr1 ∈
⋂
b∈F Nb (this intersection is
non-empty since eventually it contains a minimizing sequence for ≺2). Let b0 be the
projection (i.e., the image by the first-point map) of gr1 on [F ]. Let g1 ∈ G fixing
pointwise the branch around b0 containing gr1 and permuting the branches around b0
containing elements of F in order that the orders induced on these branches coincide
for ≺2 and g1g ≺1. The closure B1, . . . , Bn of each of these branches is homeomorphic
to D∞ itself and b0 is an end point for these subdendrites. One can now use [Duc20,
Lem. 7.14] for each bi and find an element hi ∈ Homeo(Bi) fixing b0 such that hig1g ≺1
and ≺2 coincide on F ∩Bi. Patching the hi’s (extended by the identity on the other
branches around b0), one gets an element g2 such that g2g1g ≺1 and ≺2 coincides
on F . Now, it suffices to observe that for each b ∈ F , g2g1gC1b is the branch around
g2g1gb containing the root of g2g1g ≺1 that is C2b or lies in Nb if b is the root of ≺2.
Universality. — Since G≺0 is extremely amenable, for any G-flow X, there is a G≺0-
fixed point x. The orbit map g 7→ gx is uniformly continuous and thus extends to a
G-map Ĝ/G≺0 → X. If X is moreover minimal, the image is X itself. □
Lemma 5. — Let ≺,≺′ be two convex converging linear orders with root ξ ∈ Ends(D∞)
in the G-orbit of ≺0 and F be a finite c-closed subset of Br(D∞) such that for any
x, y ∈ F , x ≺ y ⇔ x ≺′ y. Then there exists v ∈ Gξ fixing pointwise F such that
v ≺ = ≺′.
Proof. — First, we may assume that F contains the projection p of ξ to the subden-
drite 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 ≺ and ≺′.
We use a back and forth argument to construct v. Actually, we construct v as a
bijection of Br(D∞) preserving the betweenness relation and use the identification
of G with the group of betweenness preserving bijections of Br(D∞).
Let {xn}n⩾1 be an enumeration of Br(D∞)∖F . We construct increasing sequences
Xn, Yn of finite subsets of Br(D∞) containing F and bijections vn : Xn → Yn such
that for any n ∈ N:
J.É.P. — M., 2024, tome 11
Erratum to “Topological properties of Ważewski dendrite groups” 1033
(1) Xn, Yn are c-closed and contain the projection of ξ on the subdendrite they
generate.
(2) xn ∈ Xn ∩ Yn,
(3) vn preserves the betweenness relation and for any x, y ∈ Xn,
x ≺ y ⇐⇒ vn(x) ≺′ vn(y),
(4) if pn ∈ Xn is the projection of ξ on the subdendrite generated by Xn then
vn(pn) is the projection of ξ on the subdendrite generated by Yn,
(5) vn+1|Xn = vn.
We set X0 = Y0 = F and v0 to be the identity on F . We proceed by induc-
tion. Assume for some n ∈ N, Xn, Yn and vn have been defined and satisfy the
above properties. Let cn be the projection of xn+1 on the subdendrite generated by
Xn ∪ {ξ}. Let [an, bn] be the minimal arc (maybe reduced to cn if cn ∈ Xn) such that
an, bn ∈ Xn ∪ {ξ} containing cn. Let c′n be a branch point in (vn(an), vn(bn)) (with
the convention that vn(ξ) = ξ) or c′n = vn(an) = vn(bn) if cn ∈ Xn.
Let us observe that for any f ∈ Xn, cn ≺ f ⇔ c′n ≺′ vn(f). Actually, let dn =
c(ξ, cn, f) ∈ Xn, if dn = cn then cn ≺ dn, if dn = f then f ≺ cn (by the converging
property) and otherwise
f ≺ cn ⇐⇒ f ≺ an ⇐⇒ f ≺ bn
(see [Duc20, Lem. 7.5]). These properties are true because cn ∈ [an, bn] and thus, the
same holds for any c′n ∈ [vn(an), vn(bn)].
If xn = cn then let x′n be c′n and otherwise choose a point x′n in some component
not containing ξ, vn(an), vn(bn) around c′n such that for any x ∈ Xn,
xn+1 ≺ x ⇐⇒ x′n ≺′ vn(x).
This is possible because ≺′ is in the G-orbit of ≺0 and thus the linear order induced
by ≺′ on branches around c′n not containing ξ is isomorphic to the standard order
on rational numbers. Let Y ′n = Yn ∪ {x′n, c′n} and X ′n = Xn ∪ {xn+1, cn} and let us
extend vn on X ′n with vn(xn+1) = x′n and vn(cn) = c′n.
Similarly let d′n be the projection of xn+1 on the subdendrite generated by Y ′n ∪ {ξ}.
If d′n /∈ Y ′n, let [a′n, b′n] be the minimal arc such that a′n, b′n ∈ Y ′n ∪ {ξ} and contain-
ing d′n. Let dn be a branch point in
(
v−1n (a
′
n), v
−1
n (b
′
n)
)
and choose a point yn in some
component around dn not intersecting X ′n ∪ {ξ} and such that for any x ∈ Xn,
yn ≺ x ⇐⇒ xn+1 ≺′ vn(x).
Now let Xn+1 = X ′n ∪ {dn, yn}, Yn+1 = Y ′n ∪ {d′n, xn+1} and vn+1(x) = vn(x) for
all x ∈ X ′n, vn+1(dn) = d′n and vn+1(yn) = xn+1 (if these new points do not already
belong to X ′n).
By construction, Xn, Yn and vn have the announced properties and finally, for
x ∈ Br(D∞) we define v(x) = vn(x) for any n large enough such that x ∈ Xn.
The map v is a bijection that preserves the betweenness relation and maps ≺ to ≺′.
Because of the fourth property, v fixes ξ. Thus v fixes pointwise F ∪ {ξ}. □
J.É.P. — M., 2024, tome 11
1034 B. Duchesne
Proof of Theorem 4. — Since G acts transitively on Ends(D∞), we continue with
ξ = ξ0 the root of ≺0. The space Mξ0 is closed and thus compact since π1 is contin-
uous. Let π be the projection M → CCLO(D∞). This map is continuous on M and
injective on Mξ0 (in that case, the second component is constant equal to C0) with im-
age CCLO(D∞)ξ0 . Since CCLO(D∞)ξ0 and Mξ0 are both compact, these two spaces
are homeomorphic as Gξ0-flows. The first part of the proof of Theorem 2 also proves
that Gξ0/G≺0 is bi-uniformly embedded in Mξ0 and since it is dense ̂Gξ0/G≺0 ≃ Mξ0 .
Minimality and universality go as in the proof of Theorem 2. □
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/
J.É.P. — M., 2024, tome 11


Erratum to “Topological properties of Ważewski dendrite groups”

Marc Arnaudon

Koléhè Coulibaly-Pasquier

Laurent Miclo

Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds

Stéphane Fischler

Tanguy Rivoal

Values of $E$-functions are not Liouville numbers

Not a member yet

Wasserstein geometry and Ricci curvature bounds for Poisson spaces

On the existence of gr-semistable filtrations of orthogonal/symplectic $\lambda$ -connections

A non-parametric Plateau problem with partial free boundary

Moduli spaces of marked branched projective structures on surfaces

Hyperbola method on toric varieties

Mass concentration in rescaled first order integral functionals

An analytic viewpoint on the Hasse principle

Erratum to “Topological properties of Ważewski dendrite groups”

Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds

Values of $E$ -functions are not Liouville numbers

312

324