UNIQUE ERGODICITY OF THE AUTOMORPHISM GROUP OF THE SEMIGENERIC DIRECTED GRAPH

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UNIQUE ERGODICITY OF THE AUTOMORPHISM GROUP OF THE SEMIGENERIC DIRECTED

GRAPH

Colin Jahel

To cite this version:

Colin Jahel. UNIQUE ERGODICITY OF THE AUTOMORPHISM GROUP OF THE SEMI-

GENERIC DIRECTED GRAPH. 2021. �hal-03176483�

UNIQUE ERGODICITY OF THE AUTOMORPHISM GROUP OF THE SEMIGENERIC DIRECTED GRAPH

COLIN JAHEL

Abstract. We prove that the automorphism group of the semigeneric directed graph (in the sense of Cherlin’s classification) is uniquely ergodic.

1. Introduction

One key notion in the study of dynamical properties of Polish groups is amenability. A topological group is amenable when every flow, i.e. continuous action on a compact space, admits a Borel probability measure that is invariant under the action of the group.

In recent years, the study of non-locally compact Polish groups has exhibited several re- finements of this phenomenon. One of them is extreme amenability: a topological group is extremely amenable when every flow admits a fixed point (see [KPT05]). Another one is unique ergodicity: a topological group is uniquely ergodic if every minimal flow, i.e. a flow where every orbit is dense, admits a unique Borel probability measure that is invariant under the action of the group. In this paper, all measures will be Borel probability measures.

Of course, extreme amenability implies unique ergodicity, but the converse is not true as for instance, every compact group is uniquely ergodic. Beyond compactness, though, no example is known in the locally compact Polish case and Weiss proves in [Wei12] that there is no uniquely ergodic discrete group. In fact, it is suggested on page 5 in [AKL12] that in the setting of locally compact groups, compactness is the only way to achieve unique ergodicity. However, some examples appear in the non-locally compact Polish case. The first of these examples wasS_∞, the group of all permutations of Nequipped with the pointwise convergence topology (this was done by Glasner and Weiss in [GW02]). Angel, Kechris and Lyons then showed, using probabilistic combinatorial methods, that several groups of the form Aut(F), where F is a particular kind of countable structure called Fraïssé limit, are also uniquely ergodic (see [AKL12]).

A Fraïssé limit is a countable first-order homogeneous structure in the sense of model theory whose age, i.e. the set of its finite substructures up to isomorphism, is a Fraïssé class. A classF of finite structures is a Fraïssé class if it contains structures of arbitrarily large (finite) cardinality and satisfies the following:

(HP) IfA∈ F andB is a substructure ofA, thenB∈ F.

(JEP) IfA, B ∈ F then there existsC∈ F such thatAandB can be embedded inC.

(AP) If A, B, C∈ F andf:A→B,g:A→C are embeddings, then there existsD∈ F and h:B→D,l:C→D embeddings such thath◦f =l◦g.

Examples of Fraïssé classes include the class of finite graphs, the class of finite graphs omitting a fixed clique, the class of finiter-uniform hypergraphs for anyr∈N. The unique ergodicity of the automorphism groups of the limits of those classes was proven in [AKL12].

Date: January 2019.

2010Mathematics Subject Classification. Primary: 37B05 ; Secondary: 22F50, 03C15, 43A07.

Key words and phrases. Unique ergodicity, ergodic decomposition, semigeneric directed graph.

Research was partially supported by the ANR project AGRUME (ANR-17-CE40-0026).

1

The Fraïssé limit of a Fraïssé class is unique up to isomorphism. By definition, Fraïssé limits are homogeneous, i.e. any isomorphism between two finite parts of the structure can be extended in an automorphism of the structure. For more details on Fraïssé classes see [Hod93].

In [PSar], using methods from [AKL12], Pawliuk and Sokić extended the catalogue of uniquely ergodic automorphism groups with the automorphism groups of homogeneous directed graphs, which were all classified by Cherlin (see [Che98]), leaving as an open question only the case of the semigeneric directed graph.

This graph, which we denoteS, is the Fraïssé limit of the classSof simple, loopless, directed, finite graphs that verify the following conditions:

i) the relation⊥, defined byx⊥yiff¬(x→y∨y→x), is an equivalence relation, ii) for any x₁6=x₂, y₁6=y₂ such thatx₁⊥x₂ andy₁⊥y₂, the number of (directed) edges

from{x1, x₂} to{y1, y₂} is even,

where →denotes the directed edge. We will refer to ⊥-equivalence classes as columns and to the second condition as the parity condition. The⊥-class of an elementa∈S will be referred to asa^⊥.

x1 x1 x1

x2 x2 x2

y₁ y₁ y₁

y2 y2 y2

Figure 1. The three possible configurations (up to isomorphism) of two pairs of equivalent points respecting the parity condition.

More details on this structure will be given in the next section.

In this paper, we prove:

Theorem 1. The topological groupAut(S)is uniquely ergodic.

The method we use is different from the one found in [AKL12] and [PSar] since we do not work with the so-called "quantitative expansion property", but rather show that an ergodic measure can only take certain values on a generating part of the Borel sets. It is also different from the approach in [Tsa14] (see Theorem 7.4) which only applies when the structure eliminates imag- inaries. Our method relies on the idea that if there are equivalence classes in a structure and the universal minimal flow is essentially the convex orderings regarding the equivalence classes, then the ordering inside the equivalence classes and the ordering of the equivalence classes are independent, provided that the automorphism group behaves well enough.

Acknowledgements:

I am grateful to my PhD supervisors Lionel Nguyen Van Thé and Todor Tsankov for their helpful advice during my research on this paper. I also want to thank Miodrag Sokić for his comments on this paper. I thank the referee, whose comments helped me greatly improve the structure of the paper.

2. Preliminaries

The starting point of our proof is common with that of [AKL12]: to prove that Aut(S) is uniquely ergodic, it suffices to show that one particular action is uniquely ergodic, namely, its universal minimal flow, Aut(S) y M (Aut(S)). This is the unique minimal Aut(S)-flow that maps onto any minimal Aut(S)-flow (such a flow exists for any Hausdorff topological group by a classical result of Ellis, see [Ell69]); an explicit description was made by Jasiński, Laflamme,

Nguyen Van Thé and Woodrow in [JLNW14]. It is the space of expansions ofSwhose Age is a certain classS^∗.

Before describing this class, we give some more background onS. Observe that the parity condition is equivalent to the fact for everyA∈ S and two columnsP, QinA∈ S, we have for allx, x⁰ ∈P,

(∀y∈Q ((x→y)⇔(x⁰→y))) or (∀y∈Q ((x→y)⇔(y→x⁰))). This remark allows us to define the equivalence relation∼_Q onP as:

x∼Qx⁰⇔ ∀y∈Q(x→y⇔x⁰ →y).

Note that as a consequence of the parity condition, we get that inS,

∀y∈Q(x→y⇔x⁰ →y)⇔ ∃y∈Q (x→y andx⁰→y).

We can now consider P⁰ and P¹ the two ∼_Q equivalence classes in P, and we have P = P⁰tP¹. Note that each of these class could be empty. Similarly, we haveQ=Q⁰tQ¹, where Q⁰ and Q¹ are ∼P-equivalence classes. Note that at that stage, this labelling of these classes is arbitrary, which is crucial to the construction and understanding of S^∗ bellow. Indeed, the language ofS^∗has a binary relationRwhich interpretation is mainly to give a proper labelling of those equivalence classes.

This description has an interesting consequence when we recall that there must be an edge between any two points of P and Q. Denote Pⁱ → Q^j to mean for all x∈ Pⁱ and y ∈ Q^j, we have x→ y. ThenPⁱ →Q^j, implies that Q^j → P¹⁻ⁱ, P¹⁻ⁱ →Q^1−j and Q^1−j →Pⁱ.In particular, this means that for eachi∈ {0,1}, there is a uniquej∈ {0,1} such thatPⁱ→Q^j.

Pⁱ

P¹⁻ⁱ

Q^j

Q^1−j

The classS^∗ is the class of finite structures in the languageL= (→, <, R), verifying : (A) S_|→^∗ =S,

(B) <is interpreted as a linear ordering convex with respect to the columns, i.e. the columns are intervals for the ordering. For two columnsP, Q, we will therefore write P < Qto mean that for allx∈P, y∈Qwe havex < y.

(C) ForA^∗∈ S^∗, the binary relationR^A∗ verifies (a) For allxandy⊥y⁰,R^A∗(x, y)⇔R^A∗(x, y⁰).

(b) For P, Q any two columns ofA^∗ with P < Q, there is a unique ∼Q-equivalence classu(possibly empty) inP such that

∀x∈P, y∈Q R^A∗(x, y)⇔x∈u.

(c) ForP, Q any two columns of A^∗ withP < Q, if there isx₁∈P such that for all y∈Q,R^A∗(x1, y), then

∀y∈Q, x∈P R^A∗(y, x)⇔y→x1.

And if there is no suchx₁ then we have

∀y∈Q, x∈P ¬R^A∗(y, x).

(d) Ifx⊥^A∗ y, we have¬R^A∗(x, y).

Observe that in a structureA^∗∈ S^∗,R^A∗ gives us a proper labelling of the ∼Q-equivalence classes inP whenP < Q. In particular, we can render the arbitrary decompositionP =P⁰tP¹, Q=Q⁰tQ¹ canonical by setting

x∈P¹⇔(∀y∈Q R^A∗(x, y)) and

y∈Q¹⇔(∀x∈P R^A∗(y, x)).

A remarkable property of this decomposition is that the edge relation is actually entirely defined by it. Indeed, take two columnsP, QinA^∗that we decompose as above inP=P⁰tP¹, Q=Q⁰tQ¹. We know, by construction ofRonQ, thatQ¹→P¹. As we observed before, this means thatP¹→Q⁰,P⁰→Q¹andQ⁰→P⁰.

Another point of view on this expansion is given in [JLNW14]. TakeA∈ S withncolumns P1, . . . , Pnand an expansionA^∗∈ S^∗. The expansionA^∗is interdefinable with a structureA^∗∗

in the language {→, <, Li,f} where Li,f is a unary predicate for all i ∈ {1, . . . , n} = [n] and f ∈2^[n]\i. We haveA^∗_|→,<=A^∗∗_|→,<. Assuming thatP1<^A∗. . . <^A∗Pn, then we define

L^A_i,f^∗∗ ={x∈P_i : ∀j ∈[n]\i, y∈P_j (f(j) = 1⇔R^A∗(x, y)}.

Denote M ⊂ {0,1}^S2 × {0,1}^S2 the space of expansions of S whose Age is exactlyS^∗. We will denoteE= (<^E, R^E) the elements ofM, by identification with the structure that can be inferred from the expansion. The result shown in [JLNW14] is:

Theorem A. The universal minimal flow of Aut(S)isAut(S)yM.

We are interested in showing that the Aut(S)-invariant measures onMare all equal. A useful tool of measure theory is the following Lemma (see [Gut05] Theorem 3.5)

Lemma 2. Let µ and ν be two probability measures defined on a σ-field E. If there is a family (An)_n∈N ∈ E^N stable under intersection that generates E and such that for all n ∈ N, µ(An) =ν(An), thenµ=ν.

The rest of this section is devoted to describing a familyP of clopen sets that generate the Borel sets ofM. The sets of our familyP are of the form

U_(x

i)ⁿ_i=1,(ε^j_i)1≤i<j≤n∩V_(a1 1,...,a¹_i

1),...,(a^k₁,...,a^k

ik)⊂ M.

They are defined as follows.

Let (x_i)ⁿ_i=1 be in different columns. Let (ε^j_i)_i<j≤n∈ {0,1}(ⁿ₂). An elementE= (<^E, R^E)∈ Mbelongs toU_(x

i)ⁿ_i=1,(ε^j_i)_1≤i<j≤n iff the following conditions are satisfied : (A) (x^⊥₁ <^E. . . <^Ex^⊥_n)

(B) fork < l,

R^E(xk, xl)⇔(xk→xl)^εlk.

where for allx, y ∈S andε∈ {0,1}, (x→y)^ε means (x→y) if ε= 1 and ¬(x→ y) otherwise.

The rest of R on those columns can be recovered from this by construction of S^∗. Indeed, observe that for allx∈x^⊥_k,y∈x^⊥_l , we have

R^E(x, y)⇔ x∼^S_x⊥

l

xk andR^E(xk, xl) or

x^S_x^⊥

l

xk and¬R^E(xk, xl) and

R^E(y, x)⇔ y→xk andR^E(xk, y)

or xk→yand¬R^E(xk, y) .

An important remark is that if we have a different family (x⁰₁, . . . , x⁰_n) such thatxi⊥x⁰_i, then there is a family (α^j_i)1≤i<j≤n such that

U_(x

i)ⁿ_i=1,(ε^j_i)1≤i<j≤n=U_(x0

i)ⁿ_i=1,(α^j_i)1≤i<j≤n. This can be achieved by takingα^j_i =ε^j_i ifx_i∼_x⊥

j x⁰_i andα^j_i = 1−ε^j_i otherwise.

An additional remark that will be useful throughout the paper is that for a given family (x1, . . . , xn) of elements taken in different columns,

M= G

σ∈S_n,(ε^j_i)1≤i<j≤n

U_(x

σ(i))ⁿ_i=1,(ε^j_i)_1≤i<j≤n.

We also define V_(a1

1,...,a¹_i

1),...,(a^k₁,...,a^k

ik)={E∈ M:

(a¹₁<^E · · ·<^Ea¹_i1)∧ · · · ∧(a^k₁ <^E· · ·<^Ea^k_ik)}

where (a^j_i ⊥a^j_i0⁰) iffj=j⁰.

This collection of sets is a generating family for the open sets of our space, so it is also a generating family for the Borel sets.

To use Lemma 2, we would also need to know that this family is stable under intersection, unfortunately this is not the case. However, the intersection of two sets inPis actually a disjoint union of sets inP. Therefore if we considerP⁰ the collection of finite intersection of elements of P, the evaluation of a measure on an element of P⁰ is determined by the evaluation of the measure onP. By Lemma 2, any measure is entirely characterized by its evaluation on elements ofP⁰, so it is characterized by its evaluation on elements ofP.

3. Invariant measures

From this point on, we denoteG= Aut(S). Let us first define µ₀ a G-invariant probability measure onM. We defineµ₀ by:

µ0

U_(x

i)ⁿ_i=1,(ε^j_i)1≤i<j≤n∩V_(a1 1,...,a¹_i

1),...,(a^k₁,...,a^k

ik)

= 1

n!2(ⁿ²) 1

k

Y

j=1

ij! .

We call µ0 the uniform measure. It is proven in [PSar] that this measure is well-defined on all Borel sets and that it is G-invariant. We want to show that it is actually the only invari- ant measure. By Lemma 2, we only have to check that the invariant measures coincide on U_(x

i)ⁿ_i=1,(ε^j_i)1≤i<j≤n∩V_(a1 1,...,a¹_i

1),...,(a^k₁,...,a^k

ik).

Before proving Theorem 1 we need to prove the following preliminary results:

Proposition 3. For all(xi)ⁿ_i=1 such that¬(xi⊥xj) fori6=j and(ε^j_i)_i<j≤n ∈2(ⁿ²), we have:

µ U_(x

i)ⁿ_i=1,(ε^j_i)1≤i<j≤n

= 1

n!2(ⁿ²). Proposition 4. For all(a¹₁, . . . , a¹_i

1, . . . , a^k₁, . . . , a^k_i

k)such thata^j_i ⊥a^j_i0⁰ iffj=j⁰, we have:

µ V_(a1

1,...,a¹_i

1),...,(a^k₁,...,a^k

ik)

= 1

k

Y

j=1

ij! .

Similar results were proven in [PSar]. We will prove those results using different methods.

The proof of Proposition 4 is very similar to what we will do later on in order to conclude and contains the key argument of this paper.

For proofs of Proposition 4 and Theorem 1, we will need an ergodic decomposition theorem, thus we need to define the notion of ergodicity.

Definition. Let Γ be a Polish group acting continuously on a compact spaceX. AΓ-invariant measureν is said to be Γ-ergodicif for allA measurable such that

∀g∈Γ ν(A4g·A) = 0, we haveν(A)∈ {0,1}.

We can now state the following (see [Phe01] Proposition 12.4):

Theorem B. Let Γ be a Polish group acting continuously on a compact space X. Let P(X) denote the space of probability measures on X andPΓ(X) ={µ∈P(X) : Γ·µ=µ}. Then, the extreme points ofPΓ(X)are theΓ-ergodic invariant measures.

We will also need to use Neumann’s Lemma (see [Cam99], Theorem 6.2) :

Theorem C. Let H be a group acting on Ωwith no finite orbit. LetΓand∆ be finite subsets ofΩ, then there is h∈H such that h·Γ∩∆ =∅.

The remaining of the section will be divided in three subsection. One for the proof of Propo- sition 3, one for the proof of Proposition 4 and finally one for the proof of Theorem 1.

3.1. Proof of Proposition 3. For this proof, we will need the following technical lemma.

Lemma 5. Let k < n, letP₁, . . . , P_n be different columns inS and let y₁ ∈P₁, . . . , y_k ∈ P_k. Take a given family ε^j_i ∈ {0,1} where 1 ≤ i < j ≤ n and k < j. Then there exist yk+1 ∈ Pk+1, . . . , yn ∈Pn such that(yi→yj) iffε^j_i = 1 for alli < j andk < j.

Proof. Takex_k+1∈P_k+1, . . . , x_n∈P_n. Consider the following structure A= ((y^A₁, . . . , y^A_n, x^A_k+1, . . . , x^A_n),→^A)

where (y_i^A→^A y^A_j)⇔(yi →yj) ifi < j≤k, (y_i^A→^Ay_j^A) ⇐⇒ (ε^j_i = 1) if 1≤i < j ≤nand k < j. We also havex^A_i ⊥^Ay_i^A fori > k and (x^A_i →^Ax^A_j ⇔xi→xj) fork < i < j.

We put edges betweenx^A_i andy_j^Ain order for them to respect the parity condition. Remark that there is more than one way to do this, for instance one can ask that when k < i < j, (x^A_i →^A y^A_j)⇔(x^A_i →^A x^A_j) and (x^A_j →^A y^A_i )⇔(y_j^A →^A y_i^A). The remaining edges can be added arbitrarily because they concern columns with only one vertex.

We make sure thatA∈ S. Indeed, noting that since there is one point in the firstkcolumns, and two in the remaining ones, it suffices to check the parity condition in the lastn−kcolumns.

Takek < j < i≤n. We know that the edges betweenx^A_i andy_j^Aand the edge betweenx^A_i and x^A_j go in the same direction. Similarly, the edge betweenx^A_j andy_i^A and the edge between y^A_j andy_i^Aalso go in the same direction. Therefore the parity condition must be respected.

Remark that ((y^A₁, . . . , y^A_k, x^A_k+1, . . . , x^A_n),→^A) and ((y1, . . . , yk, xk+1, . . . , xn),→^S) are iso- morphic, henceAembeds inSin a way that extends this isomorphism. The image of (y^A_k+1, . . . , y^A_n) is as wanted.

The fundamental observation for the proof of Proposition 3 is that if we takex₁, . . . , x_n ∈S all in different columns,

Aut(S)·(<^∗, R^∗) = G

σ∈S_n,(ε^j_i)1≤i<j≤n

U_(x

σ(i))ⁿ_i=1,(ε^j_i)_1≤i<j≤n.

We will show that for any two familiesε= ε^j_i

i<j≤n,α= α^j_i

i<j≤n andσ∈Sn there is a g∈Gsuch that

U_(xi)ⁿ

i=1,ε=g·U_(xσ(i))ⁿ

i=1,α.

This means that all sets of this form have the same measure, hence we will have the result because there aren!2(ⁿ²) such sets.

First, we constructg⁰∈Gsuch that g⁰·U_(xσ(i))ⁿ

i=1,α=U_x1,...,xn,β for someβ = (β^j_i)1≤i<j≤n.

We want to prove that there isg⁰∈Gsuch thatg⁰·xi∈(x_σ(i))^⊥. By Lemma 5, there exists x⁰₁, . . . , x⁰_n∈Ssuch thatx_σ(i)⊥x⁰_iandxi→xj iffx⁰_i→x⁰_j. Remark that by construction, there is a partial automorphismτthat sendsxσ(i)tox⁰_i. By homogeneity, there isg⁰ an automorphism ofSthat extendsτ. We remark that

g⁰·U(x_i)ⁿ_i=1,α=U(x⁰

σ(i))ⁿ_i=1,α

and as we observed before, U_(x⁰

σ(i))ⁿ_i=1,α does not depend on x⁰_i, but on their columns. Thus, there exist a familyβ= (β_i^j)_1≤i<j≤n such that

U_(x⁰

σ(i))ⁿ_i=1,α =U_(xi)ⁿ

i=1,β. Next, we constructh∈Gsuch that

U_(xi)ⁿ

i=1,ε =h·U_(xi)ⁿ

i=1,β.

Assume that there are k < l such that β_i^j =ε^j_i for all (i, j)6= (k, l) andβ_k^l 6=ε^l_k. Remark that taking care of this case will be enough to prove the result : Ifα andβ disagree in more than one coordinate, iterating this process still allows to modify coordinates one at a time.

Let us take x⁰_k ⊥ xk such that for all i ∈ [n]\{k, l}, x⁰_k → xi iff xk → xi and x⁰_k → xl

iff xl → xk. This is possible using Lemma 5 where {y1, . . . , y_n−1} = {x1, . . . , xn}\{xk} and Pn =x^⊥_k. We definex⁰_l⊥xl similarly.

We takeh ∈G such thath(xi) = xi for all i ∈[n]\{k, l}, h(x⁰_k) = xk and h(x⁰_l) =xl. By homogeneity, such ahexists: indeed, by the parity condition, we have (xk →xl)⇔(x⁰_k →x⁰_l).

Let us prove thathgives the result.

TakeE∈U_x1,...,xn,β. We will prove that

h·E∈U(x_i)ⁿ_i=1,ε. For alli < j we want to prove that

R^h·E(xi, xj)⇔(xi→xj)^εji, and since

R^h·E(x_i, x_j)⇔R^E(h⁻¹(x_i), h⁻¹(x_j)), we prove

R^E(h⁻¹(xi), h⁻¹(xj))⇔(xi→xj)^εji. If{i, j} ∩ {k, l}=∅, the result is obvious.

Ifj=kandi < k, we have:

R^h·E(xi, xk)⇔R^E(h⁻¹(xi), h⁻¹(xk))

⇔(xi→h⁻¹(xk))^βki

⇔(xi→x⁰_k)^βki

⇔(xi→xk)^βki, and sinceβ_i^k=ε^k_i, we have

R^h·E(xi, xk)⇔(xi→xk)^εki.

The other cases where|{i, j} ∩ {k, l}|= 1 are similar.

Finally, if (i, j) = (k, l), we have:

R^h·E(x_k, x_l)⇔R^E(h⁻¹(x_k), h⁻¹(x_l))

⇔(x_k→h⁻¹(x_l))^βlk

⇔(x_k→x⁰_l)^βkl

⇔(x_k→x_l)^εlk.

The last equivalence is a direct consequence of the definition of x⁰_l and the fact that β^l_k =

(1−ε^l_k).

3.2. Proof of Proposition 4. We prove the result by induction on the number kof columns.

By homogeneity, for any column (a^j₁)^⊥ andσ∈Si_j there existsg∈Gsuch that g·V_(aj

1,...,a^j_ij)=V_(aj σ(1),...,a^j_σ(

ij)), thus

µ

V_(aj 1,...,a^j_ij)

= 1 ij!. This proves the initial case.

Let us now assume that for all (a¹₁, . . . , a¹_i1, . . . , a^k−1₁ , . . . , a^k−1_ik−1) such thata^j_i ⊥a^j_i0⁰ iffj=j⁰, we have

µ

V_(a1 1,...,a¹_i

1),...,(a^k−1₁ ,...,a^k−1

ik−1)

= 1

k−1

Y

j=1

ij! .

We consider (a^k₁, . . . , a^k_ik) all in the same column and not in any (aⁱ₁)^⊥ for i < k. Remark that

V_(a1 1,...,a¹_i

1),...,(a^k₁,...,a^k

ik)=V_(a1 1,...,a¹_i

1),...,(a^k−1₁ ,...,a^k−1

ik−1)∩V_(ak 1,...,a^k

ik).

We want to prove that the ordering of (a^k₁)^⊥ is independent from the ordering of the other columns.

Enumerate as (V₁, . . . , Vτ) all the different sets of the formV_(a1 σ1 (1),...,a¹

σ1 (i1 )),...,(a^k−1

σk−1 (1),...,a^k−1

σk−1 (ik−1 ))

whereσ_j is a permutation of{1, . . . , i_j}. Thusτ=

k−1

Y

j=1

i_j!.

For alll∈ {1, . . . , τ}, we define

µ_Vl(·) = µ(· ∩V_l) µ(Vl) .

This is the conditional probability ofµgivenVl. We remark that:

µ=

τ

X

l=1

µ(V_l)µ_Vl.

Denote LO((a^k₁)^⊥) the space of linear orderings on (a^k₁)^⊥. There is a restriction maprfrom Mto LO((a^k₁)^⊥). We denoteV_(a^rk

1,...,a^k_ik)the image ofV_(ak 1,...,a^k

ik)byr. Letνbe, the pushforward ofµon LO(a¹₁^⊥) byr, and letνV_l be the pushforward ofµV_l by the same map. We have:

ν =

τ

X

l=1

µ(Vl)νV_l.

Observe that the initial step of the induction implies thatνis the uniform measure on LO((a^k₁)^⊥)

We denote Stab^set_(ak

1)^⊥the setwise stabilizer of (a^k₁)^⊥, Stab^pw

(a¹₁,...,a¹_i

1,...,a^k−1₁ ,...,a^k−1_ik−1)the pointwise stabilizer of (a¹₁, . . . , a¹_i

1, . . . , a^k−1₁ , . . . , a^k−1_i

k−1) and setH= Stab^set_(ak

1)^⊥∩Stab^pw

(a¹₁,...,a¹_i

1,...,a^k−1₁ ,...,a^k−1_ik−1). We remark thatνV_l isH-invariant for all l∈ {1, . . . , τ}.

Since LO(a¹₁^⊥) is compact, by Theorem B, if we prove thatν isH-ergodic, then we have the result. Indeed, thenν is an extreme point of theH-invariant measures and all theνV_lare equal toν, thus for any lwe have

µ V_(ak

1,...,a^k

ik)∩V_l

=µ_Vl V_(ak

1,...,a¹

ik)

µ(V_l)

=νV_l

V_(a^rk

1,...,a¹

ik)

µ(Vl)

=ν V_(a^r1

k,...,a¹

ik)

µ(V_l)

= 1 ik!

1

k−1

Y

j=1

i_j! and this equality finishes the induction.

It only remains to prove the ergodicity ofν. The following lemma will allow us to conclude.

Lemma 6. Let K be a group acting on a setN with no finite orbits. DenoteLO(N)the space of linear orderings onN. Then the uniform measureλonLO(N)isK-ergodic.

Proof. Take A a Borel subset of LO(N) such that for allg ∈K λ(A4g·A) = 0. Let ε > 0.

There is a cylinder, i.e. a set depending only on a finite set of N, B = B(b1, . . . , bk) such that µ(B4A) ≤ ε. Using Neumann’s Lemma, we get that there exists g ∈ K such that {b1, . . . , bk} ∩g· {b1, . . . , bk}=∅.

Moreover, since ν is uniform, the orderings of two disjoint sets of points are independent.

Indeed, taking (a₁, . . . , a_i) and (c₁, . . . , c_i⁰) two disjoint families of points. Note that λ(a₁ <

· · · < a_i∩c₁ < · · · < c_i⁰) is equal to the number of way to insert (c₁, . . . , c_i⁰) in (a₁, . . . , a_i) respecting both orderings times the weight of a given ordering of (a₁, . . . , a_i, c₁, . . . , c_i⁰). We therefore have

λ(a₁<· · ·< a_i∩c₁<· · ·< c_i⁰) = i+i⁰

i

1 (i+i⁰)!

= 1 i!

1 i⁰!. This means thatB andg·B are independent. We can now write:

λ(A)−λ(A)² =

λ(A∩g·A)−λ(A)²

≤ |λ(A∩g·A)−λ(B∩g·A)|+|λ(B∩g·A)−λ(B∩g·B)|

+

λ(B∩g·B)−λ(B)² +

λ(B)²−ν(A)²

≤4ε.

The last inequality comes from the following inequalities

|λ(A∩g·A)−λ(B∩g·A)| ≤λ((A4B)∩g·A)≤ε,

|λ(B∩g·A)−λ(B∩g·B)| ≤λ(g·(A4B)∩B)≤ε, λ(B∩g·B) =λ(B)²

and

λ(B)²−λ(A)²

= (λ(A) +λ(B))|λ(A)−λ(B)| ≤2ε.

This proves thatλisK-ergodic.

We only have to prove that H has no finite orbits on (a¹₁)^⊥. It suffices to remark that for alla∈S, (u1, . . . , ui)∈S, there are infinitely manyb∈a^⊥ such that a→uj iffb→uj for all 1≤j≤i.

Indeed, take k ∈ N. Consider the structure ((a1, . . . , ak, v1, . . . , vi),→), where al ⊥ aj, al →vk iffa →uk and vm →vm⁰ iffum →um⁰ for all l, j ≤k and m, m⁰ ≤i. It is obvious that this structure verifies the parity condition. Therefore inSwe can findk copies ofa in its column for anyk >0.

This is enough to conclude thatν is indeedH-ergodic.

3.3. Proof of Theorem 1. In what follows, we will show that µ(U∩V) =µ(U)µ(V) for allU =U_(x

i)ⁿ_i=1,(ε^j_i)1≤i<j≤n andV =V_(a1 1,...,a¹_i

1),...,(a^k₁,...,a^k

ik). It will follow thatµ=µ0. Let us take a certain set{x₁, . . . , x_n}where none of thex_iare in the same column. We denote m the number of sets U as above associated to this family. We consider (U_i)^m_i=1 the disjoint sets of M corresponding to the ways of defining a relation R and an order on the columns x^⊥₁, . . . , x^⊥_n, i.e. Ui=U_(xσ(i))ⁿ

i=1,ε for some σ∈Sn andε∈2(ⁿ²). Proposition 3 tells us that:

∀i, j∈ {1, . . . , m}, µ(Ui) =µ(Uj).

We remark that this quantity is _m¹. We now define, for alli∈ {1, . . . , m}, µU_i(·) = µ(· ∩Ui)

µ(Ui) .

This is the conditional probability ofµgivenUi. DenoteH the subgroup ofGthat stabilizes x^⊥_i for all 1≤i≤nand each∼_x⊥

j -equivalence class inx^⊥_i fori6=j. Remark thatH stabilizes Ui, by construction, henceµU_I isH-invariant.

A simple but fundamental remark is that since

m

G

i=1

Ui = M and all the Ui have the same measure underµ, we have

µ= 1 m

m

X

i=1

µU_i.

Let LO_p(S) denote the space of partial orders that are total on each column and do not compare elements of different columns. There is a restriction map from M to LO_p(S). We considerλthe pushfoward ofµon LO_p(S) by this map. Similarly, we considerλ_Uithe pushfoward ofµU_i on LOp(S). We have

λ= 1 m

m

X

i=1

λU_i.

The rest of the proof is similar to the proof of Proposition 4: we prove thatλisH-ergodic.

TakeAa Borel subset of LO_p(S) such that for allh∈H,λ(A4h·A) = 0. For anyε >0, there is a cylinder B that depends only on finitely many points (b₁, . . . , b_k) such that λ(A4B)≤ε.

We now want to find an elementg∈H such thatB andg·B areλ-independent.

Take {b₁, . . . , b_k} ⊂ S. Remark that there is{b⁰₁, ..., b⁰_k} ⊂ S disjoint from {b₁, . . . , b_k} such that bl⊥b⁰_l andbl∼x_j b⁰_l for all 1≤l≤k and 1≤j≤n. Therefore there is an element of H that sends{b1, . . . , bk}to{b1, . . . , bk} and is therefore as wanted.

Just as in the proof of Proposition 4, we have:

λ(A)−λ(A)² =

λ(A∩g·A)−λ(A)²

≤ |λ(A∩g·A)−λ(B∩g·A)|+|λ(B∩g·A)−λ(B∩g·B)|

+

λ(B∩g·B)−λ(B)² +

λ(B)²−λ(A)²

≤4ε.

Thusλ(A)∈ {0,1}.

Since LOp(S) is compact, we have the result: λ is an extreme point of the H-invariant measures and all theλU_i are equal. Therefore we have,

µ(V ∩U_i) =µ_Ui(V)µ(U_i)

=λU_i(V)µ(Ui)

=λ(V)µ(Ui)

=µ(V)µ(U_i) for alli∈ {1, . . . , m}, andV =V_(a1

1,...,a¹_i

1),...,(a^k₁,...,a^k

ik). This finishes the proof of Theorem 1.

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Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche Email address: [email protected]