Je Remercie Aussi

María Nathalie Aubrun

Isabel Cortez

Tatiana Nagnibeda

Quentin

Christopher

Colin Thomas

Camille Et Marie

Lucille

Thomas

Keywords:

Je voudrais exprimer ma sincère gratitude à Damien Gaboriau, mon directeur de thèse. Merci de m'avoir partagé tant de belles mathématiques, d'avoir orienté mes recherches de manière décisive durant ma thèse, tout en me laissant une liberté considérable. Je te remercie d'avoir été à l'initiative d'une multitude de rencontres mathématiques qui m'ont permis de me sentir pleinement intégré dans notre communauté mathématique.

les Champclauses, et bien d'autres choses encore.

Merci à ma famille, qui a su me donner le goût du savoir et des sciences. L'épigraphe de cette thèse vous est dédié.

Axelle, merci pour tout le bonheur que tu m'apportes. Quant à Sirius, bienvenue à toi /

Table of contents

Introductions

Part I Allostery

Chapter 1: Continuum of allosteric actions for non-amenable surface groups . . . . . . . .

Part II Quantitative orbit equivalence for Z-actions

Introductions Introduction en français

La théorie des systèmes dynamiques étudie le comportement d'un espace sur lequel un groupe de transformations agit. Cette thèse se concentre sur deux branches importantes de cette théorie : la dynamique topologique et la théorie ergodique des actions de groupes. La première consiste en l'étude d'actions par homéomorphismes sur des espaces compacts. La seconde consiste en l'étude des actions de groupes sur des espaces mesurés qui admettent des mesures invariantes.

Ce manuscript est découpé en trois parties indépendantes. Dans cette introduction, qui se veut illustrative, nous expliquons les principales notions étudiées au cours de cette thèse et détaillons les principaux résultats obtenus. µ est une mesure de probabilité borélienne Γ-invariante sur C, qui est ergodique (les ensembles mesurables Γ-invariants sont de mesure 0 ou 1).

Les actions minimales ergodiques apparaissent naturellement dans différents contextes. Par exemple, soit Γ un sous-groupe dense d'un groupe compact G et soit µ la mesure de probabilité de Haar sur G. Alors l'action par multiplication à gauche Γ (G, µ) est une action minimale ergodique. Concrètement, on peut penser à une rotation irrationnelle x → x + θ (mod 1) qui fournit une action minimale ergodique de Z sur le tore T muni de la mesure de Lebesgue. Les actions profinies fournissent elles aussi des actions minimales ergodiques. Soit Γ un groupe dénombrable et soit Γ ≥ Γ 1 ≥ • • • ≥ Γ n ≥ . . . une suite décroissante de sous-groupes d'indice fini. Le groupe Γ agit sur chacun des ensembles finis Γ/Γ n et donc sur la limite profinie lim ← -Γ/Γ n des quotients Γ/Γ n . Cette action est une action minimale lorsque l'on munit la limite profinie de la topologie profinie. Par ailleurs, il existe une unique mesure de probabilité µ sur la limite profinie qui soit invariante par cette action. Alors l'action Γ (lim ← -Γ/Γ n , µ) est une action minimale ergodique qui est appelée action profinie. Les groupes qui admettent des actions profinies fidèles sont les groupes résiduellement finis, c'est-à-dire les groupes Γ qui possèdent une suite décroissante

Γ ≥ Γ 1 ≥ • • • ≥ Γ n ≥ . . . de sous-groupes d'indice fini telle que n≥1 Γ n = {1 Γ }.
En fait, tout groupe dénombrable admet des actions minimales ergodiques qui sont fidèles. Plus précisément, on a le résultat suivant.

Théorème ([Ele21]). -Tout groupe dénombrable Γ admet une action minimale ergodique qui est libre.

Ici, une action est libre si tout point a un stabilisateur trivial. Dans la suite, ce sont plutôt les actions non-libres qui vont nous intéresser. Un des intérêts de telles actions réside dans le fait qu'elles fournissent des familles intéressantes de sous-groupes qui sont donnés par les stabilisateurs des points. De façon équivalente, lorsque le groupe est de type fini, cela fournit des familles de graphes de Schreier du groupe, qui sont des objets d'études fondamentaux en théorie spectrale des graphes. On pourra consulter l'article de survol [Gri11] qui illustre l'utilisation de telles familles de graphes de Schreier.

Notions de liberté

Plusieurs notions de liberté générique existent. Une action minimale ergodique Γ (C, µ) est topologiquement libre si l'ensemble des points de C dont le stabilisateur est trivial est G δ dense.

essentiellement libre si l'ensemble des points de C dont le stabilisateur est trivial est de mesure pleine.

Ces deux propriétés sont des propriétés génériques, la première du point de vue topologique, la seconde dans un sens mesuré. En effet, un point d'un ensemble G δ dense est un point générique au sens topologique et un point d'un ensemble de mesure pleine est un point générique du point de vue de la mesure. Ces deux notions sont reliées dans le lemme qui suit. Énonçons tout d'abord une observation utile.

Fait. -Soit Γ C une action par homéomorphismes sur un compact. Alors l'ensemble des x ∈ C dont le stabilisateur est trivial est un ensemble G δ .

Preuve. Soit x ∈ C. Alors Stab(x) = {1} si et seulement si pour tout γ ∈ Γ \ {1}, on a γx = x, ce qui termine la preuve.

Ainsi, pour qu'une action minimale soit topologiquement libre, il suffit qu'il existe un point dont le stabilisateur est trivial.

Lemme. -Une action minimale ergodique essentiellement libre est topologiquement libre.

Preuve. Soit Γ (C, µ) une action minimale ergodique essentiellement libre. On sait déjà que l'ensemble des points dont le stabilisateur est trivial est un ensemble G δ . Puisque c'est aussi un ensemble de mesure pleine, il est en particulier non vide. Donc il existe x ∈ C tel que Stab(x) est trivial, ce qui conclut la preuve par minimalité de l'action.

La réciproque est fausse en général, ce qui nous amène à introduire la notion d'allostérie.

Allostérie

Définition. -Soit Γ un groupe dénombrable. Une action minimale ergodique de Γ est allostérique1 si elle est topologiquement libre mais pas essentiellement libre. Un groupe Γ est allostérique s'il admet une action minimale ergodique qui est allostérique.

Exemples de groupes non-allostériques

La notion d'allostérie est intimement liée à la dynamique sur l'espace des sousgroupes. Si Γ est un groupe dénombrable, on note Sub(Γ) l'espace des sousgroupes de Γ, sur lequel Γ agit par conjugaison. Les résultats de ce paragraphe illustrent la maxime suivante : si l'action Γ Sub(Γ) n'est pas suffisamment riche, alors Γ n'est pas allostérique.

Proposition. -Le groupe Z n'est pas allostérique. En particulier, on peut trouver un point x ∈ C dont le stabilisateur n'est pas trivial. Puisque ce dernier est un sous-groupe non-trivial de Z, il s'agit d'un nZ pour un certain entier n ≥ 1 et donc le cardinal de l'orbite d'un tel point x est fini. Par minimalité de l'action, on en déduit que C = Orb(x) est fini et que le stabilisateur d'aucun point de C n'est trivial. L'action Z (C, µ) n'est donc pas topologiquement libre.

Proposition. -Si Sub(Γ) est dénombrable, alors Γ n'est pas allostérique.

Preuve. Soit Γ un tel groupe. Soit Γ (C, µ) une action minimale ergodique qui n'est pas essentiellement libre. Alors µ({x ∈ C : Stab(x) = {1 Γ }}) > 0. Pour tout sous-groupe Λ ≤ Γ, notons

P Λ := {x ∈ C : Stab(x) = Λ}.
La famille des P Λ pour Λ sous-groupe de Γ forme une partition de C. Par hypothèse sur Γ, cette partition n'admet qu'un nombre dénombrable de pièces. Il existe alors un sous-groupe Λ ≤ Γ tel que µ(P Λ ) > 0. Par ailleurs, pour tout γ ∈ Γ, on a γP Λ = P γΛγ -1 , ce qui implique que µ(P Λ ) = µ(P γΛγ -1 ). Donc Λ ne possède qu'un nombre fini de conjugués.

Soit x ∈ C. Par minimalité de l'action et puisque Λ n'admet qu'un nombre fini de conjugués, il existe γ ∈ Γ et une suite (x n ) n≥0 ∈ C N tels que x n → x et Stab(x n ) = γΛγ -1 pour tout n ≥ 0. On en déduit que γΛγ -1 ≤ Stab(x). Ainsi, le stabilisateur d'aucun point de C n'est trivial. Donc l'action n'est pas topologiquement libre, ce qui termine la preuve.

On trouvera dans la Figure 1.2 des exemples explicites de groupes dont l'ensemble des sous-groupes est dénombrable. On peut démontrer un résultat similaire pour les groupes qui ne possèdent pas suffisamment de sous-groupes aléatoires invariants. Soit Γ un groupe dénombrable et soit Sub(Γ) l'espace des sousgroupes de Γ. C'est un espace compact lorsqu'on l'identifie à un sous-ensemble de {0, 1} Γ , sur lequel Γ agit par conjugaison. Un sous-groupe aléatoire invariant (SAI) de Γ est une mesure de probabilité Γ-invariante sur Sub(Γ). On peut adapter la preuve ci-dessus pour démontrer qu'un groupe dont les SAI ergodiques sont tous atomiques n'est pas allostérique. On trouvera des exemples concrets de tels groupes dans la Figure 1.2.

Sous-groupes aléatoires invariants / Sous-groupes uniforméments récurrents

Dans ce paragraphe, nous expliquons plus en détail les liens entre l'allostérie et la dynamique sur l'espace des sous-groupes. Ainsi, la mesure image de µ par Stab fournit une mesure de probabilité ergodique Γ-invariante sur Sub(Γ), c'est-à-dire un SAI ergodique. Par définition, l'action Γ (C, µ) est essentiellement libre si et seulement si ce SAI ergodique est égal à la mesure δ {1 Γ } .

Les sous-groupes aléatoires invariants admettent un pendant topologique. Un sous-groupe uniformément récurrent (SUR) est un sous-ensemble fermé, minimal et Γ-invariant de Sub(Γ). À toute action minimale ergodique Γ (C, µ), on peut associer un SUR, défini comme l'unique SUR inclus dans la clôture de l'ensemble {Stab(x) : x ∈ C}, voir [GW15]. On peut montrer que ce SUR est inclus dans le support du SAI associé à l'action [Jos21, Lem. 2.2]. Par ailleurs, ce SUR est égal à {{1 Γ }} si et seulement si l'action est topologiquement libre [LBMB18, Prop. 2.7]. On obtient ainsi la caractérisation suivante de l'allostérie.

Lemme. -Une action minimale ergodique Γ (C, µ) est allostérique si et seulement si le groupe trivial {1 Γ } est strictement contenu dans le support du SAI associé à cette action.

Exemples de groupes allostériques

L'existence de groupes allostériques a été posée par Grigorchuk, Nekrashevich et Suschanskii. Plus précisément, ils posent la question suivante.

Question ([GNS00, Prob. 7.3.3]). -Existe-t il un groupe dénombrable Γ qui admet des actions profinies allostériques ?

Les premiers exemples de groupes allostériques ont été découverts par Bergeron et Gaboriau. Ils répondent de plus à la question de [GNS00].

Théorème ([BG04]). -Soient Γ et Λ deux groupes non-triviaux qui sont résiduellement finis. Alors le produit libre Γ * Λ est allostérique (sauf si Γ et Λ sont isomorphes au groupe cyclique C 2 ). Plus précisément, le groupe Γ * Λ admet des actions profinies qui sont allostériques.

Une preuve indépendante de ce résultat pour Γ = Λ = Z est donnée par Abért et Elek dans [AE12]. Dans le chapitre 1, nous nous intéressons à une autre classe de groupes : les groupes fondamentaux π 1 (Σ g ) de surfaces fermées, connexes et orientables Σ g . Ce sont des groupes à un relateur, dont une présentation est donnée par π 1 (Σ g ) := x 1 , . . . , x g , y 1 , . . . , y g | [x 1 , y 1 ] . . . [x g , y g ] = 1 , où [x, y] = xyx -1 y -1 désigne le commutateur de x et y, et g ≥ 1 est un entier. Le résultat principal du chapitre 1 est le théorème suivant.

Théorème. -Soit Σ g une surface fermée, connexe et orientable de genre g ≥ 2.

Alors π 1 (Σ g )est allostérique. Plus précisément, il admet des actions profinies qui sont allostériques.

Une construction d'actions allostériques pour F 2

Pour conclure cette introduction à la partie I de cette thèse, nous proposons une construction explicite d'actions allostériques pour le groupe libre F 2 de rang deux. Pour ce faire, on construit d'abord des actions de F 2 sur des ensembles finis avec certaines propriétés.

Pour tout entier n ≥ 0 et p ≥ 2 premier, soit

C p,n := 2n l=0 [[0, p -1]] × [[0, p -2]] l .
Un point x de C p,n s'écrit donc sous la forme (x 0 , . . . , x l ) avec

x 0 ∈ [[0, p -1]] et x i ∈ [[0, p -2]]
pour tout 1 ≤ i ≤ l. L'entier l s'appelle la profondeur de x. On définit une permutation A sur l'ensemble C p,n de la façon suivante. Soit x ∈ C p,n de profondeur l.

-Si l est pair, alors A(x) := (x 0 , . . . , x l , 0) si l < 2n, x si l = 2n.

-Si l est impair, alors A(x) := (x 0 , . . . , x l + 1) si x l < p -2, (x 0 , . . . , x l-1 ) sinon.

Puis, on définit une autre permutation B sur l'ensemble C p,n de la façon suivante. Soit x ∈ C p,n de profondeur l.

-Si l est impair, alors B(x) := (x 0 , . . . , x l , 0).

-Si l = 0, alors B(x) := x 0 + 1 (mod p). Si l est pair > 0, alors B(x) := (x 0 , . . . , x l + 1) si x l < p -2, (x 0 , . . . , x l-1 ) sinon. Puisque (p -2)/(p -1) → 1 lorsque p → +∞, on peut fixer une suite de nombres premiers deux à deux distincts (p k ) k≥0 et une suite de nombres entiers (n k ) k≥0 qui tend vers +∞, telles que

+∞ ∏ k=0 |∂C p k ,n k | |C p k ,n k | > 0.
stabilisateur trivial est dense dans X. Puisque c'est un ensemble G δ , on en déduit que F 2 C est topologiquement libre. Par ailleurs, par définition de la mesure µ, on a

µ({x ∈ C : a ∈ Stab(x)}) = +∞ ∏ k=0 |∂C p k ,n k | |C p k ,n k | > 0.
Donc F 2 (C, µ) n'est pas essentiellement libre, ce qui implique que c'est une action allostérique.

Quelques groupes allostériques/non-allostériques

Groupe

Allostérie Raison ou Référence Réseaux de SL n (R) n ≥ 3 Les SAI ergodiques sont atomiques [SZ94] Réseaux de PSL n (Q p ) n ≥ 2, p premier Les SAI ergodiques sont atomiques [PT16] Figure 1.2. -Une liste de quelques groupes allostériques et non-allostériques. Cette liste n'est pas exhaustive. En particulier, les deux dernières lignes restent valides en plus grande généralité.

Γ * Λ ( = C 2 * C 2 ) avec Γ, Λ = {1} et résiduellement fini [BG04] π 1 (Σ g ), g ≥ 2 [Jos21]
Partie II : Équivalence orbitale quantitative des actions de Z

Équivalence orbitale

Le sujet principal de cette seconde partie est celui de l'équivalence orbitale des actions de groupes, avec un accent sur les actions de Z. Soit Γ un groupe dénombrable. Une action p.m.p. Γ (X, µ) sur un espace de probabilité (X, µ) est une action par bijections bimesurables sur X qui préservent la mesure µ. L'action de γ ∈ Γ sur x ∈ X est alors notée γ • x. Lorsqu'il est nécessaire de donner un nom à l'action, on écrira Γ α (X, µ) et l'action de γ ∈ Γ sur x ∈ X est notée α(γ)x.

Soient Γ et Λ deux groupes dénombrables. Deux actions p.m.p. Γ (X, µ) et Λ (Y, ν) sont orbitalement équivalentes s'il existe une équivalence orbitale, c'est-à-dire une bijection bimesurable Φ : X → Y telle que

Φ * µ = ν et pour µ-presque tout x ∈ X, Φ(Γ • x) = Λ • Φ(x).

Exemple d'une équivalence orbitale

Soit X := {0, 1} N et µ := ( 1 2 (δ 0 + δ 1 )) ⊗N . L'odomètre dyadique est la transformation T : X → X donnée par T(1, . . . , 1, 0, x k+1 , x k+2 , . . . ) = (0, . . . , 0, 1, x k+1 , x k+2 , . . . ) et T(1, 1, . . . ) = (0, 0, . . . ). Cette transformation est une bijection bimesurable qui préserve la mesure µ. Ainsi, les itérés de T fournissent une action p.m.p. Z (X, µ). On définit deux transformations T pair : X → X et T impair : X → X, appelées respectivement odomètre dyadique pair et odomètre dyadique impair, de la façon suivante T pair (x 0 , x 1 , . . . ) = (y 0 , y 1 , . . . ), T impair (x 0 , x 1 , . . . ) = (z 0 , z 1 , . . . ), où pour tout n ≥ 0, x 2n+1 = y 2n+1 , x 2n = z 2n et (y 0 , y 2 , . . . ) = T(x 0 , x 2 , . . . ), (z 1 , z 3 , . . . ) = T(x 1 , x 3 , . . . ).

Les transformations T pair et T impair préservent la mesure µ et commutent. Elles définissent donc une action p.m.p. Z 2 (X, µ).

Lemme. -Ces deux actions Z (X, µ) et Z 2 (X, µ) sont orbitalement équivalentes Esquisse de preuve. Soit X l'ensemble des suites (x n ) n≥0 ∈ X telles que au moins l'une des sous-suites (x 2n ) n≥0 ou (x 2n+1 ) n≥0 soit constante à partir d'un certain rang. Alors µ(X \ X ) = 1 et pour tout x ∈ X \ X , les deux assertions suivantes sont vérifiées :

-pour tout n ∈ Z, il existe (u, v) ∈ Z 2 tel que T n (x) = (T pair ) u (T impair ) v (x).

-pour tout (u, v) ∈ Z 2 , il existe n ∈ Z tel que (T pair ) u (T impair ) v (x) = T n (x).

Cela signifie que id X est une équivalence orbitale entre ces deux actions.

Si à la place de la partition {pair} {impair}, on partitionne N en fonction du reste modulo n, m ≥ 1, on obtient de manière similaire une équivalence orbitale entre deux actions ergodiques Z n (X, µ) et Z m (X, µ).

Équivalence orbitale et invariants

Dans l'exemple précédent, on a démontré que pour tout entier n, m ≥ 1, les groupes Z n et Z m admettent des actions p.m.p. ergodiques qui sont orbitalement équivalentes. En réalité, un résultat bien plus fort est vrai.

Théorème ( [OW80]). -Les actions ergodiques de groupes moyennables infinis sont toutes orbitalement équivalentes.

Ainsi, l'équivalence orbitale ne préserve aucun des invariants géométriques que l'on peut associer aux groupes moyennables. Par exemple, la dimension d du groupe Z d n'est pas préservée. Parmi les invariants géométriques, on peut citer la fonction de croissance d'un groupe, la fonction de Følner, le cône asymptotique, etc. De même, aucun invariant ergodique des actions p.m.p. n'est en général préservé par équivalence orbitale. Parmi les invariants ergodiques, on peut citer le spectre de la représentation de Koopman, l'entropie de Kolmogorov-Sinai, etc. Ce manque d'invariant d'équivalence orbitale est l'une des motivations principales à l'introduction de raffinements quantitatifs à l'équivalence orbitale.

Équivalence orbitale quantitative

Soit Γ un groupe de type fini et soit |•| Γ la longueur des mots associée à un système fini de générateurs S Γ ⊆ Γ. Étant donnée une action p.m.p. Γ (X, µ), on peut construire une structure métrique sur (X, µ), notée d Γ , en déclarant que la d Γ -distance entre deux points x, y ∈ X dans la même Γ-orbite est égale à

(X, µ) → (Y, ν) entre deux actions p.m.p. Γ (X, µ) et Λ (Y, ν) est une équivalence orbitale L p si pour tout γ ∈ Γ et λ ∈ Λ, les applications x → d Λ (Φ(x), Φ(γ • x)) et y → d Γ (Φ -1 (y), Φ -1 (λ • y)) appartiennent respectivement à L p (X, µ) et L p (Y, ν).
Par inégalité triangulaire, il suffit de vérifier ces conditions pour γ et λ appartenant respectivement à un système fini de générateurs de Γ et de Λ. Remarquons de plus que si l'on change les systèmes finis de générateurs, les nouvelles structures métriques obtenues restent bi-lipschitziennes aux anciennes structures métriques, ce qui ne change par le caractère L p d'une équivalence orbitale. Ainsi, la notion d'équivalence orbitale L p est indépendante du choix des systèmes finis de générateurs.

On dit que deux actions p.m.p. Γ (X, µ) et Λ (Y, ν) sont L p orbitalement équivalentes (p ∈]0, +∞]) s'il existe une équivalence orbitale L p entre elles. Finalement, on dit que deux actions p.m.p. sont L <p orbitalement équivalentes s'il existe une équivalence orbitale entre elles qui est une équivalence orbitale L q pour tout q < p.

Exemple d'une équivalence orbitale L p

Soit X := {0, 1} N et µ := ( 1 2 (δ 0 + δ 1 )) ⊗N . Soit T : X → X l'odomètre dyadique et T pair , T impair les odomètres dyadiques pair et impair. Comme expliqué précédemment, les actions Z (X, µ) et Z 2 (X, µ) induites par T et le couple (T pair , T impair ) sont orbitalement équivalentes, id X : X → X étant une équivalence orbitale. Nous allons calculer explicitement le degré d'intégrabilité de cette équivalence orbitale.

Dans les calculs qui suivent, on munit Z du système fini de générateurs {±1} et Z 2 du système fini de générateurs {(±1, 0), (0, ±1)}. On note d Z et d Z 2 les structures métriques obtenues sur (X, µ).

Propriétés d'intégrabilité pour

x → d Z 2 (x, T(x)). Pour tout n ≥ 0, soit A n ⊆ X l'ensemble des x ∈ X tels que x 0 = • • • = x n-1 = 1 et x n = 0.
Les ensembles (A n ) n≥0 forment une partition de X sur laquelle on peut calculer explicitement la valeur de d Z 2 (x, T(x)).

-Si x ∈ A 2n , alors T(x) = T pair (T impair ) 1-2 n (x). -Si x ∈ A 2n+1 , alors T(x) = (T pair ) 2 n+1 -1 T impair (x).
Puisque l'action Z 2 (X, µ) est libre, on en déduit que

d Z 2 (x, T(x)) = |1 -2 n | + 1 si x ∈ A 2n , |2 n+1 -1| + 1 si x ∈ A 2n+1 .
Enfin, la mesure de A n vaut 1/2 n+1 , donc l'intégrale X d Z 2 (x, T(x)) p dµ a la même nature que ∑ n≥0 2 (p-2)n , qui converge si et seulement si p < 2.

Propriétés d'intégrabilité pour

x → d Z (x, T pair (x)). Pour tout n ≥ 0, soit B n ⊆ X l'ensemble des x ∈ X tels que x 0 = x 2 = • • • = x 2=-2 = 1 et x 2n = 0. Si x ∈ B n , alors T pair (x) = T 2 2n -(2 2n-2 +2 2n-4 +•••+1) (x) = T (2 2n+1 +1)/3 (x).
Puisque l'action Z (X, µ) est libre, on en déduit que

d Z (x, T pair (x)) = 2 2n+1 + 1 3 si x ∈ B n .
Enfin, la mesure de B n vaut 1/2 n+1 , donc l'intégrale

X d Z (x, T pair (x)) p dµ
a la même nature que ∑ n≥0 2 (2p-1)n , qui converge si et seulement si p < 1/2. Un calcul similaire donne la même conclusion pour l'intégrale

x Figure 1.3. -Le graphe de f x : Z → Z 2 avec x = 000000 * * . . .

Le retour des invariants

L'équivalence orbitale quantitative permet de retrouver plusieurs invariants qui ne sont pas capturés par l'équivalence orbitale. Par exemple, Bowen démontre que deux groupes de type fini, qui admettent des actions p.m.p. essentiellement libres qui sont L 1 orbitalement équivalentes, possèdent la même fonction de croissance [Aus16b, Appen. B]. Plusieurs autres invariants sont préservés lorsque des actions essentiellement libres sont L 1 orbitalement équivalentes, comme par exemple le cône asymptotique pour les groupes à croissance polynomiale [Aus16b], ou encore l'entropie de Kolmogorov-Sinai pour les actions p.m.p. de groupes moyennables [START_REF] Austin | Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence[END_REF]. La condition "p < 1/2" obtenue dans l'équivalence orbitale du paragraphe précédent s'explique grâce au théorème suivant.

Théorème. -Soient n, n ≥ 1 deux entiers. Soient Z n (X, µ) et Z m (Y, ν) deux actions p.m.p. essentiellement libres qui sont L p orbitalement équivalentes. Alors p ≤ min(n/m, m/n).
On pourra se rapporter à [DKLMT20, Thm. 3.2] pour un énoncé plus général. On ne sait pas s'il existe des actions p.m.p. essentiellement libres

Z n (X, µ) et Z m (Y, ν) qui sont L p orbitalement équivalentes avec p = min(n/m, m/n).

Le théorème de Belinskaya est optimal

Dans le cas d'actions de Z, l'équivalence orbitale L 1 est extrêmement rigide. Deux actions p.m.p.

Z α (X, µ) et Z β (Y, ν) sont flip-conjuguées s'il existe une bijection bimesurable Φ : X → Y telle que Φ * µ = ν et -soit Φ • α(n) = β(n) • Φ pour tout n ∈ Z, -soit Φ • α(n) = β(-n) • Φ pour tout n ∈ Z.
Théorème ( [Bel68]). -Deux actions p.m.p. ergodiques Z (X, µ) et Z (Y, ν) sont L 1 orbitalement équivalentes si et seulement si elles sont flip-conjuguées.

Parmi les différents résultats du chapitre 2, nous démontrons que le théorème de Belinskaya est optimal au sens suivant.

Théorème. -Il existe des actions p.m.p. ergodiques de Z qui sont L <1 orbitalement équivalentes mais pas flip-conjuguées.

Ce résultat est le fruit d'un travail en commun avec Carderi, Le Maître et Tessera [CJLMT22]. Le chapitre 3 est une annexe du chapitre 2 dans lequel nous traduisons un résultat probabiliste dû à Liggett [Lig02] dans le langage introduit dans le chapitre 2.

Involution non croisante

Si k, l ∈ Z, on note [[k, l]] l'intervalle de Z dont les extrémités sont k et l. Une involution P : Z → Z est non-croisante si pour tout x, y ∈ Z, soit les intervalles [[x, P(x)]] et [[y, P(y)]] sont disjoints, soit l'un contient l'autre. Cette définition a une interprétation géométrique. Pour tout x ∈ Z qui n'est pas un point fixe de P, on trace le demi-cercle dans le demi-plan supérieur, qui est perpendiculaire à l'axe horizontal et dont les extrémités sont x et P(x). Alors P est non-croisante si et seulement si aucun de ces demi-cercles ne se coupe. Soit P : Z → Z une involution non-croisante sans point fixe. Alors l'application Q : x → P(x -1) + 1 est aussi une involution non-croisante sans point fixe. On peut à nouveau représenter géométriquement cette application, en traçant les demi-cercles dans le plan inférieur cette fois-ci. Ces demi-cercles sont obtenus en partant des demi-cercles qui représentent P, puis en effectuant une translation x → x + 1 puis une symétrie par rapport à l'axe horizontal. Dans le chapitre 4, nous démontrons que la réunion des demi-cercles associés à P et Q forme une courbe connexe, comme illustré dans la Figure 1.5.

D ∞ := a, b | a 2 = b 2 = 1 .
Nous démontrons le résultat suivant.

Théorème. -Toute action p.m.p. essentiellement libre Z (X, µ) est L <1 orbitalement équivalente à une action p.m.p. du groupe diédral infini D ∞ .

À contrario, une action p.m.p. essentiellement libre Z (X, µ) est L 1 orbitalement équivalente à une action de D ∞ si et seulement si le sous-groupe 2Z n'agit pas ergodiquement sur (X, µ).

Partie III : Équivalence orbitale quantitative et graphages

Graphage L'équivalence orbitale quantitative étudie les distortions que peuvent subir les structures métriques données par les actions p.m.p. de groupes de type fini. Ici, nous introduisons une notion d'équivalence orbitale quantitative qui impose que les structures métriques ne subissent aucune distorsion, mais soient au contraire isométriques.

Soit (X, µ) un espace de probabilité. Un graphage sur (X, µ) est un graphe dont l'ensemble des sommets est X et l'ensemble des arêtes est un sous-ensemble symétrique mesurable de X × X qui vérifie la condition suivante : pour tous ensembles mesurables A, B ⊆ X,

B deg A (x)dµ = A deg B (x)dµ, ( * )
où deg E (x) désigne l'ensemble des voisins de x qui sont dans E. Les graphages sont fondamentaux en théorie des graphes limites, car ce sont les objets limites de graphes de valence uniformément bornée [Lov12,Chap. 18]. Les graphages sont aussi des objets fondamentaux dans la théorie des actions p.m.p. Ce sont par exemple les outils principaux de la théorie du coût, étudiée en détail par Gaboriau [Gab00]. Soit Γ un groupe de type fini et S Γ un système fini de générateurs, c'est-à-dire un sous-ensemble fini S Γ ⊆ Γ, qui est symétrique (S Γ = S -1 Γ ), qui ne contient pas l'élément neutre e Γ et qui engendre le groupe ( S Γ = Γ). À une action p.m.p. Γ (X, µ), on peut associer un graphage dont l'ensemble des sommets est X et l'ensemble des arêtes est

{(x, y) ∈ X × X : ∃s ∈ S, s • x = y}.
La condition ( * ) est automatiquement satisfaite puisque le groupe Γ agit sur (X, µ) en préservant la mesure. Lorsque l'action est essentiellement libre, ce graphage retient la géométrie du groupe, puisque pour µ-presque tout x ∈ X, la composante connexe du graphage qui contient x est isomorphe au graphe de Cayley du groupe. Ici, le graphe de Cayley, noté (Γ, S Γ ), d'un groupe Γ muni d'un système fini de générateurs S Γ est le graphe dont l'ensemble des sommets est Γ et l'ensemble des arêtes est

{(γ, δ) ∈ Γ × Γ : ∃s ∈ S Γ , γs = δ}.

Équivalence orbitale isométrique

Dans le chapitre 5, on étudie une notion forte d'équivalence orbitale quantitative que l'on appelle équivalence orbitale isométrique.

Définition. -Soient Γ et Λ deux groupes de type fini. Fixons S Γ et S Λ des systèmes finis de générateurs pour Γ et Λ respectivement. Deux actions p.m.p. Γ α (X, µ) et Λ β (Y, ν) sont isométriquement orbitalement équivalentes s'il existe une bijection bimesurable Φ : X → Y telle que Φ * µ = ν et pour µ-presque tout x ∈ X, l'application Φ induit une isométrie entre la composante connexe de x dans le graphage associé à α et la composante connexe de Φ(x) dans le graphage associé à β.

Contrairement aux notions d'équivalence orbitale L p , la notion d'équivalence orbitale isométrique dépend du choix du système fini de générateurs de façon primordiale, puisque le graphage associé à une action en dépend.

Rigidité et flexibilité

La notion d'équivalence orbitale isométrique renforce celle d'équivalence orbitale L ∞ . En effet, une équivalence orbitale L ∞ induit presque partout une bijection uniformément bi-lipschitzienne entre les composantes connexes des graphages associés aux actions, alors qu'une équivalence orbitale isométrique induit presque partout une isométrie.

Néanmoins, les différences entre ces notions sont notoires. Contrairement au théorème de Belinskaya, qui implique que l'équivalence orbitale L 1 pour les actions ergodiques de Z est peu ou prou triviale, l'équivalence orbitale L

Introduction in English

Dynamical systems theory is the study of the behavior of a space on which a group of transformations acts. This PhD thesis focuses on two main branches of this theory: topological dynamics and ergodic theory of group actions. The former consists in the study of group actions by homeomorphisms on compact spaces, whereas the latter consists in the study of group actions on measured spaces which admit invariant measures. This manuscript is divided into three independent parts. In this introduction, which is meant to be illustrative, we explain the main notions studied during this thesis as well as the main results that we obtained.

Part I : Allostery

Minimality and ergodicity

This first part combines topological dynamics and ergodic theory of group actions. The main object of study is actions of countable groups on compact spaces, for which there exists an invariant probability measure. More precisely, a minimal ergodic action of a countable group Γ is an action Γ (C, µ) such that -Γ C is an action by homeomorphisms on a compact Hausdorff space C which is minimal, that is, every orbit is dense.

µ is a Borel probability measure on C which is Γ-invariant and ergodic, that is, any measurable Γ-invariant has measure 0 or 1.

Minimal ergodic actions appear naturally in different contexts. For instance, let Γ be a countable dense subgroup of a compact group G and let µ be the Haar probability measure on G. Then the right multiplication action Γ (G, µ) is a minimal ergodic action. Concretely, one can think of an irrational rotation x → x + θ (mod 1), which provides a minimal ergodic action of Z on the torus T endowed with the Lebesgue measure. Profinite actions form another class of minimal ergodic actions. Let Γ be a countable group and let

Γ > Γ 1 > • • • > Γ n > .
. . be a decreasing sequence of finite index subgroups. The group Γ acts on each finite space Γ/Γ n and thus on the profinite limit lim ← -Γ/Γ n . This is a minimal action when the profinite limit is endowed with the profinite topology. Moreover, there is a unique Γ-invariant probability measure µ on the profinite limit. Then the action Γ (lim ← -Γ/Γ n , µ) is a minimal ergodic action called a profinite action. Countable groups which admit profinite faithful actions are residually finite, that is, they admit a decreasing sequence

Γ 1 ≥ • • • ≥ Γ n ≥ . . . of finite index subgroups such that n≥1 Γ n = {1 Γ }.
Actually, any countable group admits minimal ergodic actions that are faithful. More precisely, we have the following result.

Theorem ([Ele21]). -Any countable group Γ admits a minimal ergodic action which is free.

Here free means that the stabilizer of every point is trivial. In the sequel, we will mostly be interested in non-free actions. One of the interests of such actions lies in the fact that they give families of subgroups, given by the stabilizer of points. Equivalently, when the group is finitely generated, a non-free action provides a family of Schreier graphs of the groups. These are fundamental objects in spectral graph theory. We refer to the survey [Gri11] for the use of such families of Schreier graphs.

Notions of freeness

Several notions of generic freeness exist. A minimal ergodic action Γ (C, µ) is topologically free if there is a G δ dense set of points having trivial stabilizer.

essentially free if there is a set of full measure of points having trivial stabilizer.

These two properties are generic properties of the action, the first one in the topological sense, the second one in the measure sense. Indeed, a point in a G δ dense set is generic from the topological point of view and a point in a full measure set is generic from the measure point of view. We will relate these two notions in the following lemma. Let us first make a useful observation.

Fact. -Let Γ C be an action by homeomorphisms on a compact Hausdorff space. Then the set of x ∈ C whose stabilizer is trivial is a G δ set.

Proof. Let x ∈ C. Then Stab(x) = {1} if and only if for all γ ∈ Γ \ {1}, we have γx = x, which proves the fact. Therefore, it is enough to have just one point with a trivial stabilizer for a minimal action to be topologically free.

Lemma. -A minimal ergodic action which is essentially free is topologically free.

Proof. Let Γ (C, µ) be a minimal ergodic action which is essentially free. We already know that the set of points with trivial stabilizer is G δ . Since this is also a set of full measure, it is in particular nonempty. Thus, there exists x ∈ C such that Stab(x) is trivial, which finishes the proof by minimality of the action.

The converse is false in general, which brings us to introduce the notion of allostery.

Allostery

Definition. -Let Γ be a countable group. A minimal ergodic Γ-action is allosteric1 if it is topologically free but not essentially free. A group Γ is allosteric if it admits a minimal ergodic action which is allosteric.

Examples of non-allosteric groups

The notion of allostery is closely related to the dynamic on the space of subgroups. If Γ is a countable group, we denote by Sub(Γ) the space of subgroups of Γ, on which Γ acts by conjugation. The results in this paragraph illustrate the following maxim: if the action Γ Sub(Γ) is not rich enough, then Γ is not allosteric.

Proposition. -The group Z is not allosteric.

Proof. Let Z (C, µ) be a minimal ergodic action, which is not essentially free. By definition

µ({x ∈ C : Stab(x) = {0}}) > 0.
In particular, there is a point x ∈ C whose stabilizer is not the trivial group. Since it is a nontrivial subgroup of Z, it is isomorphic to nZ for some n ≥ 1 and thus the cardinal of the orbit of such a point x is finite. By minimality, we deduce that C = Orb(x) is finite and that the stabilizer of every point is nontrivial. Thus, the action Z (C, µ) is not topologically free and this concludes the proof.

Proposition. -If Sub(Γ) is countable, then Γ is not allosteric.

Proof. Let Γ be a group with only countably many subgroups. Let Γ (C, µ) be a minimal ergodic action which is not essentially free. Then we have µ({x ∈ C : Stab(x) = {1 Γ }}) > 0. For any subgroup Λ ≤ Γ, let

P Λ := {x ∈ C : Stab(x) = Λ}.
The family (P Λ ) Λ≤Γ forms a partition of C. By assumption on Γ, this partition has only countably many pieces. Thus, there is a subgroup Λ ≤ Γ such that µ(P Λ ) > 0. Moreover, for all γ ∈ Γ, we have γP Λ = P γΛγ -1 . This implies that µ(P Λ ) = µ(P γΛγ -1 ). We deduce that Λ has only finitely many conjugates.

Let x ∈ C. By minimality and since Λ has only finitely many conjugates, there is γ ∈ Γ and a sequence (x n ) n≥0 ∈ C N such that x n → x and Stab(x n ) = γΛγ -1 for all n ≥ 0. We deduce that γΛγ -1 ≤ Stab(x). Thus, the stabilizer of every point is nontrivial. Therefore, the action is not topologically free, which concludes the proof.

Explicit examples of groups which admit only countably many subgroups can be found in Figure 2.2. One can show a similar result for countable groups which have few invariant random subgroups. Let Γ be a countable group and let Sub(Γ) be the space of subgroups of Γ. This is a compact subspace of {0, 1} Γ , on which Γ acts by conjugation. An invariant random subgroup (IRS) of Γ is a Γ-invariant probability measure on Sub(Γ). The above proof can be adapted to prove that any group whose ergodic IRS are all atomic is not allosteric. Concrete examples of such groups can be found in Figure 2.2.

Invariant random subgroups / Uniformly recurrent subgroups

In this paragraph, we explain in greater detail the connections between allostery and the dynamic on the space of subgroups.

Let Γ (C, µ) be a minimal ergodic action. The map

Stab : x → {γ ∈ Γ : γ • x = x} is a measurable map C → Sub(Γ). Moreover, Stab is equivariant, because for all x ∈ C and γ ∈ Γ, Stab(γ • x) = γStab(x)γ -1 .

Examples of allosteric groups

The existence of allosteric groups was raised by Grigorchuk, Nekrashevich and Suschanskii. More precisely, they asked the following question.

Question ([GNS00, Prob. 7.3.3]). -Does there exist countable groups which admit profinite allosteric actions?

The first examples of allosteric groups were found by Bergeron and Gaboriau. They moreover answer the question of [GNS00].

Theorem ([BG04]

). -Let Γ and Λ be two nontrivial, finitely generated groups. Then the free product Γ * Λ is allosteric (unless Γ and Λ are isomorphic to the cyclic group C 2 ). More precisely, the group Γ * Λ admits profinite allosteric actions.

An independent proof of this result when Γ = Λ = Z was given by Abért and Elek in [AE12]. In Chapter 1, we are interested in another class of groups: fundamental groups π 1 (Σ g ) of closed, connected and orientable surfaces Σ g . These are one relator groups, given by a group presentation of the form π 1 (Σ g ) := x 1 , . . . , x g , y 1 , . . . ,

y g | [x 1 , y 1 ] . . . [x g , y g ] = 1 ,
where [x, y] = xyx -1 y -1 is the commutator of x and y and g ≥ 1 is an integer. The main result of Chapter 1 is the following.

Theorem. -Let Σ g be a closed, connected and orientable surface of genus g ≥ 2.

Then π 1 (Σ g ) is allosteric. More precisely, it admits profinite allosteric actions.

A construction of allosteric actions for F 2

We conclude this introduction to Part I by an explicit construction of allosteric actions for the free group F 2 of rank two. For this, we first construct actions of F 2 on finite sets with specific properties.

For all integer n ≥ 0 and prime p ≥ 2, let

C p,n := 2n l=0 [[0, p -1]] × [[0, p -2]] l .
A point x ∈ C p,n can be written as x = (x 0 , . . . , x l ) with x 0 ∈ [[0, p -1]] and

x i ∈ [[0, p -2]] for all 1 ≤ i ≤ l.
The number l is called the depth of x. We define a permutation A on the set C p,n . Let x ∈ C p,n be of depth l.

-If l is even, then A(x) := (x 0 , . . . , x l , 0) if l < 2n,

x if l = 2n.
-If l is odd, then A(x) := (x 0 , . . . , x l + 1) if x l < p -2, (x 0 , . . . , x l-1 ) otherwise.

We define another permutation B on the set C p,n . Let x ∈ C p,n be of depth l.

-If l is odd, then B(x) := (x 0 , . . . , x l , 0).

-If l = 0, then B(x) := x 0 + 1 (mod p). If l is even and > 0, then B(x) := (x 0 , . . . , x l + 1) if x l < p -2, (x 0 , . . . , x l-1 ) otherwise.

Let a and b be the two generators of F 2 . The permutations A and B induce an action of F 2 on C p,n by letting a • x := A(x) and b • x := B(x) for all x ∈ C p,n . This action is illustrated in Figure 2.1. We call boundary of C p,n the set ∂C p,n consisting of all points of depth 2n. These are exactly the set of points that are fixed by a. Moreover, an elementary computation yields

|∂C p,n | |C p,n | -→ n→+∞ p -2 p -1 .
Since (p -2)/(p -1) → 1 as p → +∞, we can fix a sequence of pairwise distinct prime numbers (p k ) k≥0 and a sequence of integers (n k ) k≥0 which tends to +∞, such that

+∞ ∏ k=0 |∂C p k ,n k | |C p k ,n k | > 0.
Endowed with the product topology, the space C := ∏ k≥0 C p k ,n k is compact. We let µ be the product of the uniform probability measure on each C p k ,n k .

Proposition 2.0.1. -The diagonal action of F 2 on C is allosteric.

Sketch of the proof. First of all, F 2 (C, µ) is a minimal ergodic action. Indeed, an application of the Chinese remainder theorem allows us to show that for all K ≥ 0, the diagonal action F 2 ∏ k≤K C p k is transitive. Thus, the cylinders of C of the form U (y 1 ,...,y K ) := {(x k ) k≥0 ∈ C : x 0 = y 0 , . . . , x K = y K } all have the same measure and intersect every orbit of the action F 2 C. This implies that -the measure µ is the unique F 2 -invariant measure on C, which is thus ergodic, -the action F 2 C is minimal.

Let us prove that F 2 (C, µ) is topologically free, but not essentially free. For all k ∈ N, fix x k ∈ C p k ,n k an element of depth 0 and let

x := (x k ) k≥0 ∈ C. By construction, we observe that Stab(x) = {1 F 2 }. Since F 2
C is minimal, we deduce that the set of points with trivial stabilizer is dense in C. Since it is also a G δ set, we deduce that F 2 C is topologically free. Moreover, by definition of the measure µ, we have

µ({x ∈ C : a ∈ Stab(x)}) = +∞ ∏ k=0 |∂C p k ,n k | |X p k ,n k | > 0.
Therefore F 2 C is not essentially free, which implies that it is an allosteric action.

Some allosteric/non-allosteric groups

Group

Allostery Reason or Reference

Γ * Λ ( = C 2 * C 2 )
with Γ, Λ = {1} and residually finite [BG04] Part II: Quantitative orbit equivalence for Z-actions

π 1 (Σ g ), g ≥ 2 [Jos21] Z Z Work in progress Γ nilpotent finitely generated |Sub(Γ)| = ℵ 0 BS(1, n) |Sub(BS(1, n))| = ℵ 0 FSym(N),

Orbit equivalence

The main topic of this second part is orbit equivalence for actions of groups, especially for actions of Z. Let Γ be a countable group. A p.m.p. action Γ (X, µ) on a probability space (X, µ) is an action by bimeasurable bijections of X which preserve the measure µ. The action of γ ∈ Γ on x ∈ X is denoted by γ • x.

When it is necessary to give a name to the action, we will write Γ α (X, µ) and the action of γ ∈ Γ on x ∈ X is denoted by α(γ) • x.

Let Γ and Λ be two countable groups. Two p.m.p. actions Γ (X, µ) and Λ (Y, ν) are orbit equivalent if there exists an orbit equivalence, that is a bimeasurable bijection Φ : X → Y such that Φ * µ = ν and for µ-almost every x ∈ X,

Φ(Γ • x) = Λ • Φ(x).

Example of an orbit equivalence

Let X := {0, 1} N and µ := ( 1 2 (δ 0 + δ 1 )) ⊗N . The dyadic odometer is the transformation T : X → X given by T(1, . . . , 1, 0, x k+1 , x k+2 , . . . ) = (0, . . . , 0, 1, x k+1 , x k+2 , . . . ) and T(1, 1, . . . ) = (0, 0, . . . ). This is a bimeasurable bijection, which preserves the measure µ. This yields a p.m.p. action Z (X, µ). Let us define two transformations T even : X → X and T odd : X → X, called the even dyadic odometer and the odd dyadic odometer. They are defined by T even (x 0 , x 1 , . . . ) = (y 0 , y 1 , . . . ), T odd (x 0 , x 1 , . . . ) = (z 0 , z 1 , . . . ), where for all n ≥ 0, x 2n+1 = y 2n+1 , x 2n = z 2n and (y 0 , y 2 , . . . ) = T(x 0 , x 2 , . . . ), (z 1 , z 3 , . . . ) = T(x 1 , x 3 , . . . ).

The transformations T even and T odd preserve the measure µ and commute. Thus they give rise to a p.m.p. action Z 2 (X, µ).

Lemma. -These two actions Z (X, µ) and Z 2 (X, µ) are orbit equivalent.

Sketch of the proof. Let X be the set of all (x n ) n≥0 ∈ X such that at least one of the subsequence (x 2n ) n≥0 is eventually constant. Then µ(X \ X ) = 1 and for all x ∈ X \ X , the following properties are true:

-for all n ∈ Z, there exists (u, v) ∈ Z 2 such that T n (x) = (T even ) u (T odd ) v (x).

-for all (u, v) ∈ Z 2 , there exists n ∈ Z such that (T even ) u (T odd ) v (x) = T n (x).

This means that id X is an orbit equivalence between these two actions.

If instead of the partition {odd} {even}, we look at the partition of N given by the remainder modulo n, m ≥ 1, then one obtains similarly two orbit equivalent ergodic actions Z n (X, µ) and Z m (X, µ).

Orbit equivalence and invariants

In the preceding example, we proved that for all n, m ≥ 1, the groups Z n and Z m admit p.m.p. ergodic actions that are orbit equivalent. Much more is true.

Theorem ([OW80]

). -Any two p.m.p. ergodic actions of any two amenable infinite groups are orbit equivalent.

Thus, orbit equivalence does not preserve any of the geometric invariants that one can associate to amenable groups. For instance, the dimension d of the group Z d is not preserved. Among the geometric invariants, one can cite the growth rate of a group, the Følner function, the asymptotic cone, etc. Similarly, in general, no ergodic invariant of p.m.p. action is preserved under orbit equivalence. Among ergodic invariants, one can cite the spectrum of the Koopman representation, Kolmogorov-Sinai entropy, etc. This lack of orbit equivalence invariants is one of the main motivations for the introduction of quantitive refinement of orbit equivalence.

Quantitative orbit equivalence

Let Γ be a finitely generated group and let |•| Γ be the word length associated with some finite generating system S Γ ⊆ Γ. Given any p.m.p. action Γ (X, µ), one can construct a metric structure on (X, µ), denoted by d Γ , by letting the d Γ -distance between two points x, y ∈ X in the same Γ-orbit be equal to

d Γ (x, y) := inf{|γ| Γ : γ • x = y}.
One of the main goals of quantitative orbit equivalence is to understand how an orbit equivalence between two p.m.p. actions can distort the associated metric structures. More precisely, let Γ and Λ be two finitely generated groups and let |•| Γ and |•| Λ be the word length associated with finite generating systems for Γ and Λ respectively.

Definition. -Let p ∈]0, +∞]. An orbit equivalence Φ : (X, µ) → (Y, ν) between two p.m.p. actions Γ (X, µ) and Λ (Y, ν) is an L p orbit equivalence if for all γ ∈ Γ and λ ∈ Λ, the maps

x → d Λ (Φ(x), Φ(γ • x)) and y → d Γ (Φ -1 (y), Φ -1 (λ • y))
belong to L p (X, µ) and L p (Y, ν) respectively. By the triangle inequality, it is enough to check these conditions for γ and λ belonging to any finite generating system of Γ and Λ respectively. Let us mention that changing the finite generating systems leads to bilipschitz metric structures, which does not impact the integrability conditions. Thus, the notion of L p orbit equivalence is independent of the choice of finite generating systems.

We say that two p.m.p. actions Γ (X, µ) and Λ (Y, ν) are L p orbit equivalent (p ∈]0, +∞]) if there exists an L p orbit equivalence between them. Finally, we say that two p.m.p. actions are L <p orbit equivalent if there exists an orbit equivalence between them which is an L q orbit equivalence for all q < p.

Example of an L p orbit equivalence

Let X := {0, 1} N and µ := ( 1 2 (δ 0 + δ 1 )) ⊗N . Let T : X → X be the dyadic odometer and T even , T odd be the even and odd dyadic odometers. The actions Z (X, µ) and Z 2 (X, µ) induced by T and the couple (T even , T odd ) are orbit equivalent, the identity map id X : X → X being an orbit equivalence. We will compute explicitly the integrability degree of this orbit equivalence.

In the computations below, we endow Z with the finite generating system {±1} and Z 2 with the finite generating system {(±1, 0), (0, ±1)}. We denote by d Z and d Z 2 the corresponding metric structures on (X, µ).

Integrability properties for

x → d Z 2 (x, T(x)). For all n ≥ 0, let A n ⊆ X be the set of x ∈ X such that x 0 = • • • = x n-1 = 1 and
x n = 0. The sets (A n ) n≥0 form a partition of X on which one can compute explicitly the value of d Z 2 (x, T(x)).

-If x ∈ A 2n , then T(x) = T even (T odd ) 1-2 n (x).

-If x ∈ A 2n+1 , then T(x) = (T even ) 2 n+1 -1 T odd (x).

Since the action Z 2 (X, µ) is free, we then deduce that

d Z 2 (x, T(x)) = |1 -2 n | + 1 if x ∈ A 2n , |2 n+1 -1| + 1 if x ∈ A 2n+1 .
Finally, the measure of A n is equal to 1/2 n+1 , thus the behavior of the integral X d Z 2 (x, T(x)) p dµ comes down to the behavior of the series ∑ n≥0 2 (p-2)n , which converges if and only if p < 2.

Integrability properties for x → d Z (x, T even (x)). For all n ≥ 0, let B n ⊆ X be the set of x ∈ X such that

x 0 = x 2 = • • • = x 2=-2 = 1 and x 2n = 0. If x ∈ B n , then T even (x) = T 2 2n -(2 2n-2 +2 2n-4 +•••+1) (x) = T (2 2n+1 +1)/3 (x).
Since the action Z (X, µ) is free, we deduce that

d Z (x, T even (x)) = 2 2n+1 + 1 3 if x ∈ B n .
Finally, the measure of B n is equal to 1/2 n+1 , thus the behavior of the integral X d Z (x, T even (x)) p dµ comes down to the behavior of the series ∑ n≥0 2 (2p-1)n , which is convergent if and only if 0 < p < 1/2. A similar computation yields a similar conclusion for the integral

X d Z (x, T odd (x)) p dµ.
Since it is enough to check the integrability conditions on the generators of the groups Z and Z 2 , we deduce that the p.m.p. actions Z (X, µ) and Z 2 (X, µ) are L <1/2 orbit equivalent.

Since these actions are free and orbit equivalent, to almost every point x ∈ X one can associate a map f x : Z → Z 2 , where f x (n) is the unique element (u, v) ∈ Z 2 such that T(x) = (T even ) u (T odd ) v (x). If one looks carefully at the graph of the map f x for a generic x ∈ X, one recognizes the Lebesgue curve, see Figure 2.3. This condition "p < 1/2" is not unconnected with the fact that a continuous curve [0, 1] → [0, 1] 2 filling the square cannot be p-Hölder with p > 1/2.

x Figure 2.3. -The graph of σ(-, x) : Z → Z 2 with x = 000000 * * . . . By looking at the partitions of N given by the remainder modulo n and m ≥ 2, we get similarly two p.m.p. actions Z n (X, µ) and Z m (X, µ) which are L <p orbit equivalent for p = min(n/m, m/n). One can find these examples (less explicitely) in the article [DKLMT20, Thm. 6.9]. We explain below why this condition on p is optimal.

Return of invariants

Quantitative orbit equivalence is one way to recover several invariants that were not captured by orbit equivalence. For instance, Bowen proved that any two finitely generated groups, which admit p.m.p. essentially free actions that are L 1 orbit equivalent, have the same growth function [Aus16b, Appen. B]. Several other invariants are preserved by L 1 orbit equivalence, such as asymptotic cones for groups of polynomial growth [Aus16b], or else Kolmogorov-Sinai entropy for p.m.p. actions of amenable groups [START_REF] Austin | Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence[END_REF].

The condition "p < 1/2" obtained in the orbit equivalence of last paragraph is explained by the following theorem.

Theorem. -Let n, n ≥ 1 be two integers. Let Z n (X, µ) and Z m (Y, ν) be two p.m.p. essentially free actions that are L p orbit equivalent. Then p ≤ min(n/m, m/n).

We refer to [DKLMT20, Thm. 3.2] for a more general statement. It is unknown whether two p.m.p. essentially free actions Z n (X, µ) and Z m (Y, ν) can be L p orbit equivalent with p = min(n/m, m/n).

Belinskaya's theorem is optimal

L 1 orbit equivalence is extremely rigid when it comes to Z-actions. Two p.m.p. actions Z α (X, µ) and Z β (Y, ν) are flip-conjugate if there exists a bimeasurable bijection Φ : X → Y such that Φ * µ = ν and

-either Φ • α(n) = β(n) • Φ for all n ∈ Z, -or Φ • α(n) = β(-n) • Φ for all n ∈ Z.

Theorem ([Bel68]

). -Two p.m.p. ergodic actions Z (X, µ) and Z (Y, ν) are L 1 orbit equivalent if and only if they are flip-conjugate.

Among the results contained in Chapter 2, we prove that Belinskaya's theorem is optimal in the following sense.

Theorem. -There exists p.m.p. ergodic actions of Z which are L <1 orbit equivalent but not flip-conjugate.

This result is the fruit of a joint work with Carderi, Le Maître and Tessera [CJLMT22]. Chapter 3 is an annex of Chapter 2 in which we translate a probabilistic result due to Liggett [Lig02] in the language introduced in Chapter 2.

Non-crossing involution

For k, l ∈ Z, let [[k, l]] be the interval of Z whose extremities are k and l. An involution P : Z → Z is non-crossing if for all x, y ∈ Z, the intervals [[x, P(x)]] and [[y, P(y)]] are either disjoint, or one contains the other. This definition has a geometric interpretation. For all x ∈ Z such that P(x) = x, we draw the circular arc in the upper half-plane whose extremities are x and P(x), which is perpendicular to the horizontal axis. Then the involution P is non-crossing if and only if none of these circular arcs intersect. Let P : Z → Z be a non-crossing involution which admits no fixed point. Then the map Q : x → P(x -1) + 1 is also a non-crossing involution without fixed point. We can again draw geometrically this involution with circular arcs, this time in the lower half plane. These circular arcs on the lower half plane are obtained by taking a reflexion across the horizontal line of the arcs on the upper half-plane, followed by a translation, see Figure 2.5.

Quantitative orbit equivalence between Z and D ∞

We use in Chapter 4 this combinatorial lemma to compare p.m.p. actions of Z and of the infinite dihedral group D ∞ up to quantitative orbit equivalence. The infinite dihedral group is the group which admits the following presentation:

D ∞ := a, b | a 2 = b 2 = 1 .
We prove the following result.

Theorem. -Any p.m.p. essentially free action Z (X, µ) is L <1 orbit equivalent to some p.m.p. action of the infinite dihedral group D ∞ . By contrast, a p.m.p. essentially free action Z (X, µ) is L 1 orbit equivalent to some p.m.p. action of the infinite dihedral group D ∞ if and only if the subgroup 2Z does not act ergodically on (X, µ).

Part III: Quantitative orbit equivalence and graphings

Graphing

Quantitative orbit equivalence studies the distortions that can appear between metric structures associated with p.m.p. actions of finitely generated groups.

Here, we introduce a notion of quantitative orbit equivalence which imposes isometries between metric structures.

Let (X, µ) be a probability space. A graphing on (X, µ) is a graph whose vertex set is X and whose edge set is a measurable symmetric subset of X × X, which satisfies the following condition: for all measurable subsets A, B ⊆ X,

B deg A (x)dµ = A deg B (x)dµ, ( * )
where deg E (x) is the set of neighbors of x which belong to E. Graphings are fundamental in graph limit theory as they are the limit objects for sequences of bounded degree graphs, see [Lov12,Part 4] for an introduction to this theory. They are also one of the main objects in the cost theory of p.m.p. actions, which was extensively studied by Gaboriau [Gab00].

Let Γ be a finitely generated group and S Γ be a finite generating system, that is a finite subset S Γ ⊆ Γ, which is symmetric (S Γ = S -1 Γ ), which does not contain the identity element e Γ ∈ Γ and which generated the group: S Γ = Γ. To any p.m.p. action Γ (X, µ), one can associate a graphing, whose vertex set is X and whose edge set is

{(x, y) ∈ X × X : ∃s ∈ S, γx = y}.
Condition ( * ) is satisfied because the group Γ acts in a p.m.p. manner on (X, µ). When the action is essentially free, this graphing retains the geometry of the group, since the connected component of µ-almost every x ∈ X is isomorphic to the Cayley graph of the group. Here, the Cayley graph (Γ, S Γ ) of a group Γ with a finite generating system S Γ is the graph whose vertex set is Γ and whose edge set is

{(γ, δ) ∈ Γ × Γ : ∃s ∈ S Γ , γs = δ}.

Isometric orbit equivalence

In Chapter 5, we study a strong notion of quantitative orbit equivalence that we call isometric orbit equivalence.

Definition. -Let Γ and Λ be two finitely generated groups. Fix S Γ and S Λ finite generating systems Γ and Λ respectively. Two p.m.p. actions Γ α (X, µ) and Λ β (Y, ν) are isometric orbit equivalent if there exists a bimeasurable bijection Φ : X → Y such that Φ * µ = ν and µ-almost every x ∈ X, the map Φ induces an isometry between the connected component of x in the graphing associated with α and the connected component of Φ(x) associated with β.

Contrary to L p orbit equivalence, the notion of isometric orbit equivalence depends heavily on the finite generating systems, because the graphing associated with a p.m.p. action depends on them.

Rigidity and flexibility

Isometric orbit equivalence is a strengthening of L ∞ orbit equivalence. Indeed, an L ∞ orbit equivalence induces almost everywhere a biLipschitz bijection between connected components of the graphings associated with the actions, whereas an isometric orbit equivalence induces almost everywhere an isometry.

Nevertheless, the differences between these notions are considerable. Recall that Belinskaya's theorem implies that L 1 orbit equivalence is more or less trivial for ergodic actions of Z. On the contrary, the theory of L 1 orbit equivalence (and even L ∞ orbit equivalence) for p.m.p. actions of Z d , d ≥ 2, is much richer. For instance, Fieldsteel and Friedman proved that given any ergodic action Z 2 α (X, µ), there exists a mixing action Z 2 β (Y, ν) such that α and β are L ∞ orbite equivalent.

We show that a similar result in the context of isometric orbit equivalence is false.

Theorem. -Let d ≥ 2 and let S be a finite generating system for Z d . Let Z d α (X, µ) be a mixing action. Then any p.m.p. action Z d (Y, ν) which is isometric orbit equivalent to α is conjugate to α.

Actually, we prove a similar rigidity result for any group Γ equipped with a finite generating system S Γ such that the automorphism group of the Cayley graph (Γ, S Γ ) is countable. A theorem due to Trofimov implies that this is indeed the case for any finite generating system S of Z d , see [MS98,Thm. 4.3].

The archetype of a group which admits a Cayley graph whose automorphism group is uncountable is the free group F d on d ≥ 2 generators. We show that the rigidity phenomenon obtained for Z d cannot be true for F d .

Theorem. -Let F d be the free group of d ≥ 2 generators x 1 , . . . , x d and let S be the finite generating system S := {x ±1 1 , . . . , x ±1 d }. Then there exists ergodic actions F d α (X, µ) and F d β (Y, ν) that are isometric orbit equivalent, such that α is mixing but β is not.

Part I

Allostery

Chapter 1

Continuum of allosteric actions for non-amenable surface groups

The content of this chapter is the same as that of the article [Jos21].

Table of contents of

Introduction

Let Γ be a countable discrete group. Let α be a minimal action of Γ on a compact Hausdorff space C. The action α is topologically free if for every non-trivial element γ ∈ Γ, the set {x ∈ C | α(γ)x = x} has empty interior. This notion of freeness can be characterized by the triviality of the URS associated with the action α as follows. Let Sub(Γ) be the space of subgroups of Γ, and let Stab α : C → Sub(Γ) be the Borel map defined by

Stab α (x) := {γ ∈ Γ | α(γ)x = x}.
Here Sub(Γ) is equipped with the topology of pointwise convergence which turns it into a compact totally disconnected topological space on which Γ acts continuously by conjugation. Glasner and Weiss proved in [GW15] that there exists a unique closed, Γ-invariant, minimal subset in the closure of {Stab α (x) | x ∈ C}, called the stabilizer Uniformly Recurrent Subgroup, stabilizer URS for short, associated with the minimal action α, that we denote by URS(α). The stabilizer URS is trivial if it is equal to {{1 Γ }}. One of the feature of the stabilizer URS associated with a minimal action α is that its triviality is equivalent to the topological freeness of α, see Lemma 1.2.1. Let (X, µ) be a standard probability measure space, and let β be a probability measure preserving (hereafter p.m.p. ) action of a countable group Γ on (X, µ). The action β is essentially free if for every non-trivial γ ∈ Γ, the set {x ∈ X | α(γ)x = x} is µ-negligible. The measurable counterpart of the stabilizer URS is the stabilizer Invariant Random Subgroup, stabilizer IRS for short, associated with β. It is defined as the Γ-invariant Borel probability measure (Stab β ) * µ on Sub(Γ), and is denoted by IRS(β). A stabilizer IRS is the prototype of an IRS, which is a Borel probability measure on Sub(Γ) that is invariant under the conjugation action of Γ. The trivial IRS is the Dirac measure at the trivial subgroup. Observe that IRS(β) is trivial if and only if β is essentially free. Abért, Glasner and Virág proved that every IRS is in fact a stabilizer IRS for some p.m.p. action, see [START_REF] Abért | Kesten's theorem for invariant random subgroups[END_REF].

An ergodic minimal action Γ (C, µ) is a minimal action of Γ on a compact Hausdorff space C together with a Γ-invariant ergodic Borel probability measure µ. Thus an ergodic minimal action has both a stabilizer URS and a stabilizer IRS. It is a classical result that the essential freeness of an ergodic minimal action implies its topological freeness, see Lemma 1.2.2. In other words, if the stabilizer IRS of an ergodic minimal action is trivial, then its stabilizer URS is trivial. The present article provides new counterexamples in the study of the converse. Definition 1.1.1. -An ergodic minimal action is allosteric1 if it is topologically free but not essentially free. A group is allosteric if it admits an allosteric action.

Main Question. -What is the class of allosteric groups? First, let us discuss examples of groups that don't belong to this class. It is the case for groups whose ergodic IRS's are all atomic, i.e., equal to the uniform measure on the set of conjugates of a finite index subgroup. Indeed, we prove in Proposition 1.2.3 that the IRS of an ergodic minimal action which is topologically free is either trivial, or has no atoms. Thus, if Sub(Γ) is countable, then Γ is not allosteric, see Corollary 1.2.4. Examples of groups with only countably many subgroups are: finitely generated nilpotent groups, more generally polycyclic groups, extensions of Noetherian groups by groups with only countably many subgroups (e.g. solvable Baumslag-Solitar groups BS(1, n)), see [START_REF] Becker | Stability and invariant random subgroups[END_REF], or Tarski monsters.

There are also groups whose ergodic IRS's are all atomic for other reasons. For instance, this is the case for lattices in simple higher rank Lie groups [SZ94], commutator subgroups of either a Higman-Thompson group or the full group of an irreducible shift of finite type [START_REF] Dudko | Finite factor representations of Higman-Thompson groups[END_REF], lattices in projective special linear group PSL n (k) over an infinite countable field k [PT16]. See also [Cre17], [CP] or [START_REF] Bekka | Character rigidity of simple algebraic groups[END_REF] for other examples of groups with few ergodic IRS's. Thus, none of these groups are allosteric, because of their lack of IRS's.

More surprisingly, there exists non-allosteric groups with plenty of ergodic IRS's, such as countable abelian groups which admit uncountably many subgroups. Indeed, if Γ is such a group, then any Borel probability measure on Sub(Γ) is an IRS, but Γ is not allosteric since any minimal Γ-action which is topologically free is actually essentially free for any invariant measure, see Remark 1.4.4. Another example is given by the group FSym(N) of finitely supported permutations on N, as well as its alternating subgroup Alt(N). They both admit a lot of ergodic IRS's, see [Ver12] and [START_REF] Thomas | Invariant random subgroups of inductive limits of finite alternating groups[END_REF]. However, an argument similar to that of Lemma 10.4 in [START_REF] Thomas | Invariant random subgroups of inductive limits of finite alternating groups[END_REF] implies that neither FSym(N) nor Alt(N) is allosteric.

Let us now discuss examples of allosteric groups. Bergeron and Gaboriau proved in [BG04] that if Γ is non-amenable and isomorphic to a free product of two non-trivial residually finite groups, then Γ is allosteric. We refer to Remark 1.2.12 for a more precise statement of their results. In [AE07], Abért and Elek independently proved that finitely generated non-abelian free groups are allosteric, and in [AE12], they proved that the free product of four copies of Z/2Z admits an allosteric action whose orbit equivalence relation is measure hyperfinite. In all [BG04], [START_REF] Abért | Non-abelian free groups admit nonessentially free actions on rooted trees[END_REF] and [AE12], the allosteric actions obtained are in fact profinite, see Section 1.2.2 for a definition. These were the first known examples answering a question of Grigorchuk, Nekrashevich and Sushchanskii in [GNS00, Problem 7.3.3] about the existence of profinite allosteric actions.

The main result of this article is to prove that non-amenable surface groups, that is fundamental groups of closed surfaces other than the sphere, the torus, the projective plane or the Klein bottle, are allosteric. More precisely, we prove the following result.

Theorem 1.1.2. -Any non-amenable surface group admits a continuum of profinite allosteric actions that are pairwise topologically and measurably non-isomorphic.

Moreover, we prove that the IRS's given by the non-isomorphic allosteric actions that we construct are pairwise distinct. We refer to Theorem 1.4.1 and Theorem 1.4.2 for a precise statement of our results. Let us mention that surface groups are known to have a large "zoo" of IRS's. For instance, Bowen, Grigorchuk and Kravchenko proved in [START_REF] Bowen | Characteristic random subgroups of geometric groups and free abelian groups of infinite rank[END_REF] that any non elementary Gromov hyperbolic group admits a continuum of IRS's which are weakly mixing when considered as dynamical systems on Sub(Γ). In an upcoming work (personal communication), Carderi, Le Maître and Gaboriau prove that non-amenable surface groups admit a continuum of IRS's whose support coincides with the perfect kernel of Γ, i.e., the largest closed subset without isolated points in Sub(Γ). However, our IRS's are drastically different from the latter ones: we show that they are not weakly mixing, and that their support is strictly smaller than the perfect kernel, see Remark 1.4.4 and Remark 1.4.5.

We develop in Section 1.2 the preliminary results needed about profinite actions and allosteric actions. In particular, we prove that allostery is invariant under commensurability. In order to build ergodic profinite allosteric actions of non-amenable surface groups, we rely on a residual property of non-amenable surface groups in order to prove in Section 1.3 that they admit special kinds of finite index subgroups. The proof of Theorem 1.1.2 is completed in Section 1.4.

Acknowledgments. I would like to thank A. Le Boudec for various conversations related to this article, as well as T. Nagnibeda for letting me know that Vershik's work implies that the group of finitely supported permutations of the integers is not allosteric. I wish to thank particularly D. Gaboriau for his constant encouragement and support, as well as for his numerous remarks on this article.

Preliminaries

Topological dynamic and URS/IRS

Let C be a compact Hausdorff space, and let α be an action by homeomorphisms of a countable discrete group Γ on C. The action α is minimal if the orbit of every x ∈ C is dense. Recall that α is topologically free if for every non-trivial element γ ∈ Γ, the closed set

Fix α (γ) := {x ∈ C | α(γ)x = x}
has empty interior. Since C is a Baire space, this is equivalent to saying that the set {x ∈ C | Stab α (x) = {1 Γ }} is meager, i.e., a countable union of nowhere dense sets.

The set Sub(Γ) of subgroups of Γ naturally identifies with a subset of {0, 1} Γ . It is closed for the product topology. Thus the induced topology on Sub(Γ) turns it into a compact totally disconnected space, on which Γ acts continuously by conjugation. A URS of Γ is a closed minimal Γ-invariant subset of Sub(Γ). The trivial URS is the URS that only contains the trivial subgroup. Recall that the stabilizer URS of a minimal action α of

Γ on C is the unique closed, Γ-invariant minimal subset in the closure of {Stab α (x) | x ∈ C}. If C 0 ⊂ C denotes the locus of continuity of Stab α : C → Sub(Γ), then one can prove that URS(α) is equal to the closure of the set {Stab α (x) | x ∈ C 0 }, see [GW15].
A proof of the following classical result can be found in [ The following lemma clarifies the relation between the stabilizer URS and the stabilizer IRS. Recall that the support of a Borel probability measure is the intersection of all closed subsets of full measure. Lemma 1.2.2. -Let α be a minimal Γ-action on a compact Hausdorff space C and µ be a Γ-invariant Borel probability measure on C. Then URS(α) is contained in the support of IRS(α). In particular, if IRS(α) is trivial, then URS(α) is trivial.

Proof. Let F be a closed subset of Sub(Γ) such that µ(Stab -1 α (F)) = 1. By min- imality of α, every non-empty open subset U of C satisfies µ(U) > 0. Thus, Stab -1 α (F) is dense in C. Let x ∈ C be a continuity point of Stab α . Let (x n ) n≥0 be a sequence of elements in Stab -1
α (F) that converges to x. Then Stab α (x) ∈ F, and we thus obtain that URS(α) ⊂ F. By definition of the support of IRS(α), this implies that URS(α) ⊂ supp(IRS(α)).

The following proposition gives a partial converse to Lemma 1.2.2. Proposition 1.2.3. -Let α be a minimal Γ-action on a compact Hausdorff space C, and µ be a Γ-invariant Borel probability measure on C. If URS(α) is trivial, then IRS(α) is either trivial or atomless.

Proof. Assume that IRS(α) has a non-trivial atom {Λ}. By invariance, the atoms {γΛγ -1 } have equal measure for all γ ∈ Γ. Thus, Λ has only finitely many conjugates. Thus, the closure in Sub(Γ) of the set {Stab α (x)

| x ∈ C} contains the finite set {γΛγ -1 | γ ∈ Γ}, which is closed, Γ-invariant and minimal. Thus, URS(α) is non-trivial.
This last result implies that the converse of Lemma 1.2.2 is actually true for groups admitting only countably many subgroups. Corollary 1.2.4. -Let α be a minimal Γ-action on a compact Hausdorff space and

µ a Γ-invariant Borel probability measure on C. If Sub(Γ) is countable, then IRS(α) is trivial iff URS(α) is trivial.
Thus, groups Γ such that Sub(Γ) is countable are not allosteric.

Profinite actions and their URS/IRS

Let Γ be a countable group. For every n ≥ 0, let α n be a Γ-action on a finite set X n , and assume that for every n ≥ 0, α n is a quotient of α n+1 , i.e., there exists a Γ-equivariant onto map q n : X n+1 X n . The inverse limit of the finite spaces X n is the space lim ← -

X n := (x n ) n≥0 ∈ ∏ n≥0 X n | ∀n ≥ 0, q n (x n+1 ) = x n .
This space is closed, thus compact, and totally disconnected in the product topology. Let α be the Γ-action by homeomorphisms on lim ← -X n defined by

α(γ)(x n ) n≥0 := (α n (γ)x n ) n≥0 .
If each X n is endowed with a Γ-invariant probability measure µ n , we let µ be the unique Borel probability measure on lim ← -X n that projects onto µ k via the canonical projection π k : lim ← -X n → X k , for every k ≥ 0. The Γ-action α preserves µ, and is called the inverse limit of the p.m.p. Γ-actions α n . A p.m.p. action of Γ is profinite if it is measurably isomorphic to an inverse limit of p.m.p. Γ-actions on finite sets. A proof of the following lemma can be found in [Gri11, Prop. 4.1]. Lemma 1.2.5. -The following are equivalent:

(i) For every n ≥ 0, α n is transitive, and µ n is the uniform measure on X n .

(ii) The action α is minimal.

(iii) The action α is µ-ergodic.

(iv) The action α is uniquely ergodic, i.e., µ is the unique Γ-invariant Borel probability measure on lim ← -X n .

With the above notations, the following lemma is useful to compute the measure of a closed subset in an inverse limit (here, no group action is involved). Lemma 1.2.6. -Let A be a closed subset of lim ← -

X n . Then A = n≥0 π -1 n (π n (A)). Thus µ(A) = lim n→+∞ µ n (π n (A)). Proof. First, A is contained in ∩ n≥0 π -1 n (π n (A)) since it is contained in each π -1 n (π n (A)). Conversely, let x be in ∩ n≥0 π -1 n (π n (A)).
For every n ≥ 0, there exists y n ∈ A such that π n (x) = π n (y n ). By compactness of A, let y ∈ A be a limit of some subsequence of (y n ) n≥0 . By definition of the product topology, for every n ≥ 0, π n (x) = π n (y), thus x = y and x belongs to A.

Let (Γ n ) n≥0 be a chain in Γ, that is an infinite decreasing sequence Γ = Γ 0 ≥ Γ 1 ≥ . . . of finite index subgroups. If X n = Γ/Γ n
and µ n is the uniform probability measure on X n , then we get a profinite action that is ergodic by Lemma 1.2.5. Conversely, any ergodic (equivalently minimal) profinite Γ-action Γ lim ← -X n is measurably isomorphic to a profinite action of the form Γ lim ← -Γ/Γ n for some chain (Γ n ) n≥0 , by fixing a point x ∈ lim ← -X n , and letting Γ n be the stabilizer of

π n (x) ∈ X n .
Lemma 1.2.7. -Let (Γ n ) n≥0 be a chain in Γ, and let α be the corresponding ergodic profinite Γ-action. Then URS(α) is trivial if and only if there exists a sequence

(γ n Γ n ) n≥0 ∈ lim ← - Γ/Γ n such that n≥0 γ n Γ n γ -1 n = {1 Γ }. Proof. For all x ∈ lim ← - Γ/Γ n , if x = (γ n Γ n ) n≥0 , then Stab α (x) = n≥0 γ n Γ n γ -1 n .
Thus, the result is a direct consequence of Lemma 1.2.1.

γ n Γ n γ -1 n = {1 Γ }. For every k ≥ 0, if π k : lim ← - Γ/Γ n → Γ/Γ k denotes the projection onto the k th coordinate, then the set {Stab α (x) | x ∈ lim ← - Γ/Γ n , π k (x) = γ k Γ k } ⊂ Sub(Γ) has positive measure for IRS(α), is contained in Sub(γ k Γ k γ -1 k ) and is invariant under the finite index subgroup Stab α k (γ k Γ k ) = γ k Γ k γ -1 k .
By ergodicity, it is a full measure set. Thus, for a.e. x ∈ lim ← -

Γ/Γ n , Stab α (x) is a subgroup of γ k Γ k γ -1 k .
Since this is true for every k ≥ 0, we conclude that IRS(α) is trivial.

Allostery and commensurability

Two groups Γ 1 and Γ 2 are commensurable if there exists finite index subgroups Λ 1 ≤ Γ 1 and Λ 2 ≤ Γ 2 such that Λ 1 is isomorphic to Λ 2 . In this section, we prove the following result.

Theorem 1.2.9. -Allostery is invariant under commensurability.

We prove Theorem 1.2.9 in two steps, by showing that allostery is inherited by finite index overgroups in Proposition 1.2.10 and by finite index subgroups in Proposition 1.2.11. Let Γ be a countable group and Λ ≤ Γ a finite index subgroup. Let α : Λ (C, µ) be an action by homeomorphisms on a compact Hausdorff space C with a Λ-invariant Borel probability measure µ on C. The group Γ acts on X × Γ trivially on the first factor and by left multiplication on the second factor. This action projects onto a Γ-action by homeomorphisms on the quotient of X × Γ by the Λ-action λ • (x, γ) = (α(λ)x, γλ), and the product of µ with the counting measure projects onto a Γ-invariant Borel probability measure. This action is the Γ-action induced by α.

Proposition 1.2.10. -Let Γ be a countable group and Λ ≤ Γ a finite index subgroup. Then the Γ-action induced by any allosteric Λ-action is allosteric.

Proof. Let α : Λ (C, µ) be an allosteric action. It is an exercise to prove that the Γ-action β induced by Λ is ergodic and minimal. Moreover, IRS(β) is non-trivial since the restriction of β to Λ is not essentially free. Finally, URS(α) is trivial, thus there exists by Lemma 1.2.1 a point x ∈ C such that Stab α (x) = {1 Λ }. Let y be the projection of (x, 1) onto the quotient (C × Γ)/Λ, then Stab β (y) = {1 Γ }. Since β is minimal, this implies by Lemma 1.2.1 that URS(β) is trivial. Thus β is allosteric.

Proposition 1.2.11. -Any finite index subgroup of an allosteric group is allosteric.

Proof. Let Λ ≤ Γ be a finite index subgroup. We recall the following two facts. If Γ (X, µ) is an ergodic action, then any Λ-invariant measurable set A ⊂ X of positive measure satisfies µ(A) ≥ 1/[Γ : Λ]. Moreover, for any Λ-invariant measurable set B ⊂ X of positive measure, there exists a Λ-invariant measurable set A ⊂ B of positive measure on which Λ acts ergodically.

Let Γ be an allosteric group, and let Λ ≤ Γ be a finite index subgroup. Let N be the normal core of Λ (the intersection of the conjugates of Λ). It is a finite index normal subgroup of Γ which is contained in Λ. We will prove that N is allosteric. Proposition 1.2.10 will then imply that Λ is allosteric. We let d = [Γ : N] and we fix γ 1 , . . . ,

γ d ∈ Γ a coset representative system for N in Γ. Let Γ α (C, µ) be an allosteric action. For all x ∈ C, we define O N (x) = {α(γ)x | γ ∈ N}. This is a closed, N-invariant subset of C. By minimality of α, for all x ∈ C, X = d i=1 O N (α(γ i )x).
Moreover, since N is normal in Γ, for all x ∈ C and γ ∈ Γ, we have O N (α(γ)x) = α(γ)O N (x). This implies that µ(O N (α(γ)x)) = µ(O N (x)) and that µ(O N (x)) > 0. Let y be a point in some closed, N-invariant and N-minimal set. Then N O N (y) is minimal. Let A ⊂ O N (y) be a N-invariant measurable set of positive measure on which N acts ergodically. Let µ A be the Borel probability measure on A induced by µ. Then N (O N (y), µ A ) is an ergodic minimal action, which is still topologically free. Let us prove that it is not essentially free. Since α is allosteric, IRS(α) is atomless, see Proposition 1.2.3. Thus, for µ-a.e. x ∈ C,

Stab α (x) is infinite. Since N has finite index in Γ, this implies that for µ-a.e. x ∈ C, Stab α (x) ∩ N is infinite. Thus N (O N (y), µ A )
is not essentially free, and thus is allosteric. Remark 1.2.12. -It is proved in [BG04, Théorème 4.1] that if Γ is isomorphic to a free product of two infinite residually finite groups, then Γ admits a continuum of profinite allosteric actions. Let Γ be a non-amenable group which is isomorphic to a free product of two non-trivial residually finite groups. Then Kurosh's theorem [Ser77, Section 5.5] implies that Γ admits a finite index subgroup Γ isomorphic to a free product of finitely many (and at least two) residually finite infinite groups. Proposition 1.2.10 then implies that Γ is allosteric.

Finite index subgroups of surface groups 1.3.1 Residual properties of surface groups

A surface group is the fundamental group of a closed connected surface. If the surface is orientable, then its fundamental group is called an orientable surface group, and a presentation is given by

x 1 , y 1 , . . . , x g , y g | [x 1 , y 1 ] . . . [x g , y g ] = 1 ,
for some g ≥ 1 called the genus of the surface (if g = 0, then the surface is a sphere, and its fundamental group is trivial). If the surface is non-orientable, we call its fundamental group a non-orientable surface group. It has a presentation given by x 1 , . . . , x g | x 2 1 . . . x 2 g = 1 , for some g ≥ 1 called the genus of the surface. A surface group is non-amenable if and only if it is the fundamental group of a surface other that the sphere, the torus (orientable surfaces of genus 0 and 1), the projective plane or the Klein bottle (non-orientable surfaces of genus 1 and 2). Definition 1.3.1. -Let p be a prime number. A group Γ is a residually finite p-group if for every non-trivial element γ ∈ Γ, there exists a normal subgroup N Γ such that Γ/N is a finite p-group and γ / ∈ N. Equivalently, Γ is a residually finite p-group if and only if there exists a chain (Γ n ) n≥0 in Γ consisting of normal subgroups such that for every n ≥ 0, the quotient Γ/Γ n is a finite pgroup, and

n≥0 Γ n = {1 Γ }.
Baumslag proved in [START_REF] Baumslag | On generalised free products[END_REF] that orientable surface groups are residually free, i.e., for every non-trivial element γ, there exists a normal subgroup N Γ such that Γ/N is a free group and γ / ∈ N. Moreover, free groups are residually finite p-groups for every prime p, a result independently proved by Takahasi [START_REF] Takahasi | Note on chain conditions in free groups[END_REF] and by Gruenberg in [START_REF] Gruenberg | Residual properties of infinite soluble groups[END_REF] (using a result of Magnus [START_REF] Magnus | Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring[END_REF]). This implies the following well-known result.

Theorem 1.3.2. -Orientable surface groups are residually finite p-groups for every prime p.

Remark 1.3.3. -By a result of Baumslag [START_REF] Baumslag | Residually free groups[END_REF], non-amenable non-orientable surface groups are also residually p-finite groups for every prime p. However, we leave as an exercise to the interested reader the fact that the fundamental group of a Klein bottle is not residually p for some prime p. We will not require these results.

Special kind of finite index subgroups in surface groups

Let A, B be two non-empty totally ordered finite sets. In what follows, when writing ∏ i∈A or ∏ j∈B we mean that the product is computed with respect to the increasing order of A or B respectively. We let Γ A,B be the group defined by the generators (a i , α i ) i∈A and (b j , β j ) j∈B , and the relation

∏ i∈A [a i , α i ] = ∏ j∈B [b j , β j ].
Then Γ A,B is isomorphic to a non-amenable orientable surface group, and every non-amenable orientable surface group is isomorphism to Γ A,B for some nonempty totally ordered finite sets A and B. The group Γ A,B naturally splits as an amalgamated product

Γ A,B = Γ A * Z Γ B
where Γ A and Γ B are the free groups of rank 2|A| and 2|B| respectively, freely generated by (a i , α i ) i∈A and (b j , β j ) j∈B respectively. If A ⊂ A and B ⊂ B, there is a natural onto group homomorphism Γ A,B Γ A ,B defined on the generators by

a i → a i for every i ∈ A , b j → b j for every j ∈ B , α i → α i for every i ∈ A , β j → β j for every j ∈ B , a i , α i → 1 for every i ∈ A \ A , b j , β j → 1 for every j ∈ B \ B .
We say that this morphism erases the generators a i , α i , b j , β j for i ∈ A \ A and j ∈ B \ B , see Figure 1.1. Algebraically, Γ A ,B is isomorphic to the quotient of Γ A,B by the normal closure of the set {(α i ,

β i ) | i ∈ A \ A } ∪ {(b j , β j ) | j ∈ B \ B } in Γ A,B
, and the homomorphism Γ A,B Γ A ,B corresponds to the quotient group homomorphism. (iv) None of the left coset x ∈ Γ/Λ is fixed by δ.

Proof. Fix A, B two non-empty totally ordered finite sets, such that Γ is isomorphic to Γ A,B . Let S be the set of generators (a i , α i ) i∈A and (b j , β j ) j∈B . Let j 0 be the smallest element in

B. Let γ ∈ Γ \ {1 Γ } and δ ∈ Γ B \ Z Γ B .
Let p be a prime number, and r ∈]0,

1[∩Z[1/p].
Step 1: Cyclic covering. Let ϕ : Γ A,B Z be the onto homomorphism defined on the generators of Γ A,B by

ϕ(b j 0 ) = 1, ϕ(β j 0 ) = 0, ϕ(a i ) = ϕ(α i ) = ϕ(b j ) = ϕ(β j ) = 0 for every i ∈ A, j ∈ B \ {j 0 }.
For every d ≥ 1, we let Λ d be the kernel of the homomorphism Γ → Z/dZ obtained by composing ϕ with the homomorphism of reduction modulo d. Then Λ d is a surface group. Let us describe a generating set for Λ d . For every 0 ≤ k ≤ d -1 and i ∈ A, let a i,k and α i,k be the conjugates of a i and α i respectively, by b k j 0 . Similarly, let b j,k , and β j,k be the conjugate of b j and β j respectively, by b k j 0 . Then Λ d is generated by the set

d-1 k=0 {a i,k , α i,k | i ∈ A} ∪ d-1 k=0 {b j,k , β j,k | j ∈ B \ {j 0 }} ∪ {b d j 0 , β j 0 }.
Γ/ Γ A Γ / / Γ B / Z Γ B , which implies that Γ B ∩ Γ A Γ is equal to Z Γ B .
This would thus imply that δ ∈ Z Γ B , a contradiction.

Step

∈ {0, . . . , d -1}, π(b k j 0 δb -k j 0 ) / ∈ N . Let us prove that the subgroup Λ := π -1 (N ) of Γ satisfies the four conclusions of the theorem. Proof of (i). Either γ / ∈ Λ d and thus γ / ∈ Λ, or γ ∈ Λ d and π(γ) / ∈ N. Proof of (ii). Since the index of N in Λ d /N is a power of p, [Λ d : Λ] is also a power of p. Thus [Γ : Λ] = [Γ : Λ d ][Λ d : Λ] is a power of p.
Proof of (iii). By construction, x ∈ Γ/Λ is fixed by every element in Γ A if and only its image under the canonical

[Λ d : Λ]-to-one map Γ/Λ → Γ/Λ d is equal to b k j 0 Λ d for some k ∈ E. Since |E| = rd, there are exactly rd[Λ d : Λ] = r[Γ : Λ] such x ∈ Γ/Λ. Proof of (iv). If δ / ∈ Λ d , then none of the coset x ∈ Γ/Λ is fixed by δ. If δ ∈ Λ d , then for all k ∈ {0, . . . , d -1}, we have π(b k j 0 δb -k j 0 ) / ∈ N , and thus δb -k j 0 Λ = b -k j 0 Λ. By normality of Λ in Λ d , we deduce that none of the coset x ∈ Γ/Λ is fixed by δ.

Proof of the main theorem

In this section, we give the proof of Theorem 1.1.2. More precisely, we prove the following results.

Theorem 1.4.1 (Orientable case). -Let Γ be a non-amenable orientable surface group, and fix a decomposition Γ = Γ A * Z Γ B as above. Let Z Γ B be the normal closure of the amalgamated subgroup Z in Γ B . Then there exists a continuum (α t ) 0<t<1 of ergodic profinite allosteric actions of Γ such that for all 0 < t < 1, 1. The set of points whose stabilizer for α t contains Γ A has measure t. In particular, for all 0 < s < t < 1, the actions α s and α t are neither topologically nor measurably isomorphic, and the probability measures IRS(α s ) and IRS(α t ) are distinct.

Theorem 1.4.2 (Non-orientable case). -Let Γ be a non-amenable non-orientable surface group. Then there exists an index two subgroup Γ ≤ Γ which is isomorphic to an orientable surface group, and which decomposes as Γ = Γ A * Z Γ B , and a continuum (β t ) 0<t<1 of ergodic profinite allosteric actions of Γ such that for all 0 < t < 1, the set of points whose stabilizer for β t contains Γ A has measure t/2. In particular, for all 0 < s < t < 1, the actions β s and β t are neither topologically nor measurably isomorphic, and the probability measures IRS(β s ) and IRS(β t ) are distinct.

During the proof of these theorems, we will need the following lemma.

Lemma 1.4.3. -Let Γ be a group, and Λ 1 , . . . , Λ n be finite index subgroups of Γ. If the indices [Γ : Λ i ], i ∈ {1, . . . , n}, are pairwise coprime integers, then the left coset action

Γ Γ/(Λ 1 ∩ • • • ∩ Λ n ) is isomorphic to the diagonal action Γ Γ/Λ 1 × • • • × Γ/Λ n of the left coset actions Γ Γ/Λ i .
Proof. The kernel of the group homomorphism

Γ → Γ/Λ 1 × • • • × Γ/Λ n defined by γ → (γΛ 1 , . . . , γΛ n ) is equal to Λ 1 ∩ • • • ∩ Λ n . Thus Γ/(Λ 1 ∩ • • • ∩ Λ n ) is isomorphic to a subgroup of Γ/Λ 1 × • • • × Γ/Λ n . Moreover, for every 1 ≤ i ≤ n, [Γ : Λ 1 ∩ • • • ∩ Λ n ] = [Γ : Λ i ][Λ i : Λ 1 ∩ • • • ∩ Λ n ],
and since the indices [Γ : Λ i ] are pairwise coprime, this implies that [Γ :

Λ 1 ∩ • • • ∩ Λ n ] is divisible by [Γ : Λ 1 ] . . . [Γ : Λ n ]. Thus, the group homomorphism Γ/Λ 1 ∩ • • • ∩ Λ n → Γ/Λ 1 × • • • × Γ/Λ n is
(x 1 , . . . , x n ) ∈ Γ/Λ t 1 × • • • × Γ/Λ t n such that (δx 1 , . . . , δx n ) = (x 1 , . . . , x n ).
If n is large enough, then this last number is zero by construction of Λ t n . Thus, Lemma 1.2.6 implies that Fix α t (δ) is µ t -negligible.

Finally, let us prove that the actions α t are not essentially free. By construction, the indices [Γ : Λ t i ] are pairwise coprime. Thus, Lemma 1.4.3 implies that the number of x ∈ Γ/Γ t n that are fixed by every element in Γ A is equal to the number of (y 1 , . . . ,

y n ) ∈ Γ/Λ t 1 × • • • × Γ/Λ t
n that are fixed for the diagonal action by every element in Γ A . By construction of Λ t i , this number is equal to

r 1 [Γ : Λ t 1 ] × • • • × r n [Γ : Λ t n ] which is equal to r 1 . . . r n [Γ : Γ t n ]
. Thus, Lemma 1.2.6 implies that the µ t -measure of the set of points whose stabilizer for α t contains Γ A is t. In particular, this implies that IRS(α t ) is non-trivial. Thus α t is allosteric. Moreover, this also implies that for all 0 < s < t < 1, the actions α s and α t are not measurably isomorphic, and thus not topologically isomorphic since every α t is uniquely ergodic by Lemma 1.2.5, and this finally implies that the measures IRS(α s ) and IRS(α t ) are distinct.

Proof of Theorem 1.4.2. Let Σ be a non-orientable surface of genus g ≥ 3. Consider the usual embedding of an orientable surface Σ of genus g -1 into R 3 in such a way that the reflexions in all 3 coordinate planes map the surface to itself, and let ι to be the fixed-point free antipodal map x → -x. Then Σ is homeomorphic to the quotient of Σ by ι, and the covering Σ → Σ/ι ≈ Σ is called the orientation covering. We decompose Σ as the union of two surfaces Σ A and Σ B with one boundary, of genus |A| and |B| respectively, with |A| ≤ |B|, so that ι(Σ A ) ⊂ Σ B . Fix a point p ∈ Σ A ∩ Σ B , then Van Kampen's Theorem implies that the fundamental group Γ of the surface Σ based at p is isomorphic to

Γ A * Z Γ B with Γ A = π 1 (Σ A , p), Γ B = π 1 (Σ B , p) and Z ≈ π 1 (Γ A ∩ Γ B , p).
The fundamental group Γ of Σ based at p = ι(p) naturally contains the subgroup Γ as an indextwo subgroup. Fix a curve contained in Σ B that joins p to ι(p). This produces an element γ 0 ∈ Γ \ Γ, that satisfies γ 0 Γ A γ -1 0 ≤ Γ B . Let (α t ) 0<t<1 be a continuum of allosteric Γ-actions on (X t , µ t ) given by Theorem 1.4.1. The actions β t : Γ (Y t , ν t ) induced by the Γ-actions α t are allosteric, see Proposition 1.2.10. Let us prove that the set of points in Y t whose stabilizer for β t contains Γ A has ν t -measure t/2. Since β t is an induced action and [Γ : Γ] = 2, the Γ -action β t is measurably isomorphic to a p.m.p. Γ -action on (X t × {0, 1}, µ t × unif), still denoted by β t , that satisfies the following two properties:

1. For every γ ∈ Γ \ Γ, the sets X t × {0} and X t × {1} are switched by β t (γ).

2. For every γ ∈ Γ, for every x ∈ X t , β t (γ)(x, 0) = (α t (γ)x, 0) and

β t (γ)(x, 1) = (α t (γ 0 γγ -1 0 )x, 1). This implies that for all (x, ε) ∈ X t × {0, 1}, the subgroup Γ A is contained in Stab β t (x, ε) if and only if either ε = 0 and Γ A is contained in Stab α t (x), or ε = 1 and γ 0 Γ A γ -1 0 is contained in Stab α t (x).
Thus, the set of points whose stabilizer for β t contains Γ A has ν t -measure

t + µ t ({x ∈ X t | γ 0 Γ A γ -1 0 ≤ Stab α t (x)}) 2 .
In order to finish the proof, it is enough to prove that the intersection of γ 0 Γ A γ -1 0 and Γ B \ Z Γ B is non-trivial, since any element in Γ B \ Z Γ B acts essentially freely for α t . The conjugation by γ 0 induces a group automorphism ϕ :

Γ → Γ, such that ϕ(Γ A ) ≤ Γ B . Since Γ A is not contained in the derived subgroup D(Γ), so is ϕ(Γ A ). But the amalgamated subgroup Z is contained in D(Γ), thus so is Z Γ B . This implies that the intersection ϕ(Γ A ) ∩ (Γ B \ Z Γ B ) is non- empty.
We deduce that the set of points whose stabilizer for β t contains Γ A has ν t -measure t/2. We conclude that the actions β t are neither measurably nor topologically pairwise isomorphic and that their IRS are pairwise disjoint as in Theorem 1. This implies that the support of IRS(α) doesn't contain any non-trivial subgroup with only finitely many conjugates, because otherwise the closure of the set {Stab α (x) | x ∈ C} would contain a closed minimal Γ-invariant set = {{1 Γ }}.

Chapter 1

Carderi, Gaboriau and Le Maître proved (personal communication) that the perfect kernel of a surface group coincides with the set of its infinite index subgroups. This implies that allosteric actions of surface groups are not totipotent (a p.m.p. action is totipotent if the support of its IRS coincide with the perfect kernel of the group, see [START_REF] Carderi | On dense totipotent free subgroups in full groups[END_REF]).

Remark 1.4.5. -A p.m.p. action Γ (X, µ) is weakly mixing if for every ε > 0 and every finite collection Ω of measurable subsets of X, there exists a γ ∈ Γ such that for every

A, B ∈ Ω |µ(γA ∩ B) -µ(A)µ(B)| < ε.
With this definition, it is easily seen that the restriction of a weakly mixing action to a finite index subgroup remains weakly mixing. Thus Proposition 1.2.8 implies that the IRS's of non-amenable surface groups we have constructed are not weakly mixing.

Remark 1.4.6. -The proof of our main theorem applies mutatis mutandis to branched orientable surface groups, that is fundamental groups of closed orientable branched surfaces (see Figure 1.3). These groups can be written as amalgams. Fix an integer g ≥ 2 as well as 2g letters x 1 , y 1 , . . . , x g , y g . Fix a partition of {1, . . . , g} into n nonempty intervals A 1 , . . . , A n . Let Γ k be the free group generated by x i and y i for every i ∈ A k , and let Z → Γ k be the injective homomorphism defined by sending the generator of Z to the product ∏ i∈A k [x i , y i ]. Then the amalgam * Z Γ i is a branched orientable surface group, and any branched orientable surface group can be obtained this way. The fundamental group of a closed orientable branched surface of genus ≥ 2 is a residually p-finite group for every prime p, see [KM93, Theorem 4.2]. Thus our method of proof applies to branched orientable surface groups, with any Γ k in the role played by Γ A during the proof of Theorem 1.4.1. Belinskaya's theorem is optimal

The content of this chapter is the same as that of the article [CJLMT22], which is a joint work with A. Carderi, F. Le Maître and R. Tessera.

Introduction

Given two ergodic measure-preserving (invertible) transformations T 1 , T 2 of a standard probability space (X, µ), the conjugacy problem asks whether there is a third measure-preserving invertible transformation S such that ST 1 = T 2 S.

Although the conjugacy problem is intractable in full generality, various invariants have been devised over the years. Two of the most important ones are the spectrum and the dynamical entropy. The first completely classifies compact transformations [HvN42], while the second completely classifies Bernoulli shifts [START_REF] Yakov | On the Notion of Entropy of a Dynamical System[END_REF]HvN42].

In this paper, we are interested in natural weakenings of the conjugacy problem obtained through the notion of orbit equivalence. Two measure-preserving transformations T 1 , T 2 are orbit equivalent if there is a measure-preserving transformation S such that ST 1 S -1 and T 2 have the same orbits (such an S is called an orbit equivalence between T 1 and T 2 ). A stunning theorem of Dye states that all ergodic measure-preserving transformations of a standard probability space are orbit equivalent [START_REF] Henry | On Groups of Measure Preserving Transformations[END_REF], so orbit equivalence for measure-preserving ergodic transformations is a trivial weakening of conjugacy.

In order to circumvent this indistinguishability, we will compare orbit equivalences between measure-preserving transformations in a quantitative way. This fits into the emerging field of quantitative orbit equivalence for group actions. One of its tacit aims is to capture meaningful geometric invariants, such as Følner functions [DKLMT20], growth rates [Aus16b], etc., or ergodic theoretic invariants, such as dynamical entropy [START_REF] Austin | Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence[END_REF].

In our setup of measure-preserving transformations, quantifications will be imposed on orbit equivalence cocycles. Given an orbit equivalence S between two ergodic measure-preserving transformations T 1 and T 2 , the orbit equivalence cocycles c 1 , c 2 : X → Z are the maps uniquely defined by the following equation:

for all x ∈ X ST 1 (x) = T c 2 (x) 2 S(x) and T 2 S(x) = ST c 1 (x) 1 (x).
(2.1)

Belinskaya's theorem is probably the first result on quantitative orbit equivalence. In the literature, it is often stated as a symmetric result on integrable orbit equivalence of ergodic measure-preserving transformations. However, her result is asymmetric and can be stated as follows.

Theorem 2.1.1 (Belinskaya [Bel68]). -Let T 1 and T 2 be two ergodic measurepreserving transformations, let S be an orbit equivalence between them and suppose that the previously defined cocycle c 1 is integrable, i.e. X |c 1 (x)| dµ < +∞.

Then T 1 and T 2 are flip-conjugate: either T 1 is conjugate to T

2 or T -1 1 is conjugate to T 2 .
It is natural to wonder whether Belinskaya's theorem remains valid if one weakens the integrability assumptions. For example, one could ask that one of the orbit equivalence cocycle belongs to L p (X, µ) for some p ∈ (0, 1).

We will consider more general integrability assumptions. Given a function ϕ : R + → R + , we say that a measurable integer-valued function

f is ϕ-integrable if X ϕ(| f (x)|)dµ < +∞.
Our first main result concerns orbit equivalence of measure-preserving transformations for which one of the orbit equivalence cocycles is ϕ-integrable, for some sublinear function ϕ, that is satisfying lim t→+∞ ϕ(t)/t = 0. This is for example the case for ϕ(t) = t p . With this integrability condition, the conclusion of Belinskaya's theorem does not hold.

Theorem 2.1.2 (see Theorem 2.4.14). -Let ϕ : R + → R + be a sublinear function and T 1 be an ergodic measure-preserving transformation. Then there is an ergodic measure-preserving transformation T 2 and an orbit equivalence S between T 1 and T 2 such that the associated cocycle c 1 is ϕ-integrable but the transformations T 1 and T 2 are not flip-conjugate.

The fact that the hypotheses on ϕ are fairly weak gives us much freedom. For example, the above theorem even implies that Belinskaya's theorem does not hold if we assume that one of the two orbit equivalence cocycles belongs to L p (X, µ) for all p ∈ (0, 1). Indeed if we consider for instance the sublinear function ϕ(t) = t/ ln(t + 1), then ϕ-integrability implies being in L p (X, µ) for all p < 1.

A symmetric way to strengthen Theorem 2.1.2 involves the concept of ϕintegrable orbit equivalence. We say that two measure-preserving transformations are ϕ-integrable orbit equivalent if there is an orbit equivalence S such that both orbit equivalence cocycles c 1 and c 2 are ϕ-integrable. In this context, we obtain a similar conclusion to Theorem 2.1.2, but we have to make one additional assumption on T 1 .

Theorem 2.1.3 (see Corollary 2.3.11). -Let ϕ : R + → R + be a sublinear function. Let T 1 be an ergodic measure-preserving transformation and assume that (T 1 ) n is ergodic for some n ≥ 2. Then there is another ergodic measure-preserving transformation T 2 such that T 1 and T 2 are ϕ-integrable orbit equivalent but not flip-conjugate.

Concrete examples of transformations to which this theorem applies are

Bernoulli shifts, irrationnal rotations on the circle and the m-odometer for any integer m. One can show that the only ergodic measure-preserving transformations that are not covered by this theorem are the ones that factor onto the universal odometer, that is, the transformation t → t + 1 on the profinite completion Z.

Let us point out that the proof of Theorem 2.1.2 uses Theorem 2.1.3, so the two results are not independent. As we will explain later, Theorem 2.1.2 also depends on the Baire category theorem.

However, Theorem 2.1.3 is somewhat more explicit. It relies on the following simple construction, which was already used in [LM18, Thm. 4.8]. We begin with an ergodic transformation T 1 with (T 1 ) n ergodic. We also need a periodic transformation P all of whose orbits have cardinality n and are contained in those of T 1 . The transformation T 2 is constructed by composing P and the transformation induced by T 1 on a fundamental domain of P. Then T 1 and T 2 have the same orbits. However, (T 2 ) n is not ergodic and thus T 1 and T 2 are not flip-conjugate. As a byproduct, the transformations T 1 and T 2 do not have the same spectrum, as the spectrum of T 2 contains exp(2iπ/n) whereas the spectrum of T 1 doesn't. The main task then becomes to construct P so that the orbit equivalence cocycles between T 1 and T 2 satisfy the required integrability conditions.

For many concrete measure-preserving transformations, explicit examples of such periodic transformations P with specific integrability conditions can be obtained. We will give details in the case of the Bernoulli shift, see Example 2.3.3.

Shannon orbit equivalence and dynamical entropy

A remarkable consequence of Theorem 2.1.3 can be stated in the context of Shannon orbit equivalence, as defined by Kerr and Li [START_REF] Kerr | Entropy, Shannon orbit equivalence, and sparse connectivity[END_REF]. Two measure-preserving transformations are Shannon orbit equivalent if there exists an orbit equivalence between them whose orbit equivalence cocycles c 1 and c 2 have both finite Shannon entropy. After showing that dynamical entropy is an invariant of Shannon orbit equivalence for measure-preserving actions of many groups, such as Z n for every n ≥ 2, they implicitly asked whether dynamical entropy is an invariant of Shannon orbit equivalence for measure-preserving transformations and wondered whether Shannon orbit equivalence could actually directly imply flip-conjugacy. We show that it is not the case. Theorem 2.1.4 (see Theorem 2.3.17). -Let T 1 ∈ Aut(X, µ) be an ergodic transformation and assume that (T 1 ) n is ergodic for some n ≥ 2. Then there exists T 2 ∈ Aut(X, µ) such that T 1 and T 2 are Shannon orbit equivalent but not flip-conjugate.

The above theorem is obtained by applying Theorem 2.1.3 with any sublinear function ϕ such that ln(1 + t) = O(ϕ(t)). Indeed, for any such function, ϕintegrable orbit equivalence implies Shannon orbit equivalence, see Theorem 2.3.15.

We also observe that Shannon orbit equivalence preserves finiteness of dynamical entropy, see Proposition 2.3.20. This is now subsumed by a recent preprint of Kerr and Li who proved that the dynamical entropy is preserved under Shannon orbit equivalence [START_REF] Kerr | Entropy, virtual abelianness, and shannon orbit equivalence[END_REF].

Question 2.1.5. -For which unbounded sublinear metric-compatible functions ϕ is it true that dynamical entropy is an invariant of ϕ-integrable orbit equivalence?

By the above discussion, we already know that this holds for all ϕ such that ln(1 + t) = O(ϕ(t)). On the other hand, using Dye's theorem, it is not hard to see that any two ergodic measure-preserving transformations are ϕintegrable orbit equivalent for some sublinear unbounded function ϕ (cf. the proof of [DKLMT20, Prop. 4.24]). So not every sublinear unbounded function satisfies the condition of the question.

ϕ-integrable full groups

The proof of both our main results will make crucial use of the notion of ϕ-integrable full group. Whenever T is an ergodic measure preserving transformation of the probability space (X, µ), Dye defined a Polish group [T], called the full group of T. This group is by definition the set of all measure-preserving transformations U of (X, µ) whose orbits are contained in T-orbits. More precisely, U ∈ [T] if there is a function c U , called the T-cocycle of U, such that U(x) = T c U (x) (x) for all x ∈ X. The above stated theorem of Dye, that all ergodic transformations are obit equivalent, was originally stated in terms of full groups: whenever T 1 and T 2 are ergodic transformations, the full groups [T 1 ] and [T 2 ] are conjugate.

In our context, once ϕ is fixed, the reasonable analogue of the full group associated with this integrability condition, would be the set of transformations U ∈ [T] such that the cocycle c U is ϕ-integrable. However, for this set to be a subgroup of [T], we will have to impose a mild restriction on ϕ. We say that ϕ : R + → R + is a metric-compatible function if

• (subadditivity) for all s, t ∈ R + , ϕ(s + t) ≤ ϕ(s) + ϕ(t).
• (separation) ϕ(0) = 0 and ϕ(t) > 0 for all t > 0.

• (monotonicity) ϕ is a non-decreasing function.

The name metric-compatible comes from the observation that whenever d is a metric and ϕ a metric-compatible function, then ϕ • d is also a metric. The following theorem is a combination of Lemma 2.2.14 and Theorem 2.4.1.

Theorem 2.1.6. -Let ϕ be a metric-compatible function and let T be a measure preserving transformation of the probability space (X, µ). Then the set

[T] ϕ := U ∈ [T] : X ϕ(|c u (x)|)dµ < +∞ is a group. Moreover the function d ϕ,T (U, V) := X ϕ(|c U (x) -c V (x)|)dµ
is a complete, right-invariant and separable metric on [T] ϕ whose induced topology is a group-topology. In particular ([T] ϕ , d ϕ,T ) is a Polish group.

It turns out that any sublinear function is dominated by a sublinear metriccompatible function, see Lemma 2.2.12. This will allows us to reduce the proof of Theorem 2.1.2 and Theorem 2.1.3 to the case where ϕ is metric-compatible and thereby to exploit the group structure of [T] ϕ .

Genericity of weakly mixing

Let us come back to Theorem 2.1.2. As the conclusions of Theorem 2.1.3 are stronger, we just need to show Theorem 2.1.2 whenever (T 1 ) n is non-ergodic for all n ≥ 2. Observe that this condition is incompatible with the notion of weakly mixing, as all the powers of any weakly mixing transformation are ergodic. Therefore our strategy is to provide for every ergodic transformation T 1 a weakly mixing transformation T 2 which has the same orbits as T 1 and whose T 1 -cocycle is ϕ-integrable. We do not have any constructive argument for this and we proceed through the Baire category theorem.

Theorem 2.1.7 (see Theorem 2.4.15). -Let ϕ be a sublinear metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic element. Then the set of all measurepreserving transformations in [T] ϕ which are weakly mixing and have the same orbits as T is a dense G δ set in the Polish space of aperiodic transformations of [T] ϕ .

Besides the Baire category theorem, there are two other main ingredients in the proof of Theorem 2.1.7. One is a result of Conze [START_REF] Pierre | Équations fonctionnelles et systèmes induits en théorie ergodique[END_REF] which claims that starting from any ergodic mesure-preserving transformation, the first return map to a generic measurable subset gives rise to a weakly mixing transformation. The second is a sublinear ergodic theorem which may be of independent interest. Theorem 2.1.8 (see Theorem 2.4.5). -Let ϕ : R + → R + be a sublinear nondecreasing function. Let U ∈ Aut(X, µ) and f :

X → C measurable such that ϕ(| f |) ∈ L 1 (X, µ). Then for almost every x ∈ X lim n 1 n ϕ n-1 ∑ k=0 f (U k (x)) = 0.
The convergence also holds in L 1 , that is

lim n X 1 n ϕ n-1 ∑ k=0 f (U k (x)) dµ = 0.
Outline of the paper In Section 4.1, after a few preliminaries, we present the framework and establish basic properties of ϕ-integrable full groups. In Section 2.3, we explain our construction of periodic transformations in ϕ-integrable full groups and use it to prove Theorem 2.1.2. In Section 2.4, we first prove that ϕ-integrable full groups are Polish groups. We then use the Baire category theorem and prove that weakly mixing elements are generic in the set of aperiodic elements in [T] ϕ . Combining this with Theorem 2.1.2, we finally prove Theorem 2.1.3. In the appendix, we present a proof of Belinskaya's theorem which is due to Katznelson and is not publicly available to our knowledge.

Quantitative orbit equivalence and full groups

Preliminaries

Throughout the paper, (X, µ) will denote a standard probability space without atoms. Recall that such spaces are measurably isomorphic to the interval [0, 1] equipped with the Lebesgue measure. A bimeasurable bijection T : X → X is a measure-preserving transformation of (X, µ) if for all measurable sets A ⊆ X, one has µ(T -1 (A)) = µ(A). We denote by Aut(X, µ) the group of all measurepreserving transformations of (X, µ), two such transformations being identified if they coincide on a conull set. The group Aut(X, µ) will be equipped with the uniform metric d u defined by

d u (T 1 , T 2 ) := µ({x ∈ X : T 1 (x) = T 2 (x)}).
This metric is bi-invariant and complete [Hal17, p. 73].

Remark 2.2.1. -One can always modify measure-preserving transformations on null-sets without changing its equivalence class in Aut(X, µ). Indeed the saturation of any null set is still a null set. This will often be used implicitly in the sequel.

The support of a measure-preserving transformation T ∈ Aut(X, µ) is the measurable set supp(T) := {x ∈ X : T(x) = x}.

A measure-preserving transformation T ∈ Aut(X, µ) is periodic if the Torbit of almost every x ∈ X is finite. A fundamental domain of a periodic transformation T ∈ Aut(X, µ) is a measurable subset A ⊆ X which intersects almost every T-orbit at exactly one point. Every periodic transformation admits such a fundamental domain, as can be seen by fixing a Borel linear order < on X and taking for D the set of <-least points in each orbit of the transformation. A measure-preserving transformation T ∈ Aut(X, µ) is aperiodic if the T-orbit of almost every x ∈ X is infinite. It is ergodic if every T-invariant measurable set is either null or conull.

The full group of a measure-preserving transformation T is the group Two measure-preserving transformations T 1 , T 2 ∈ Aut(X, µ) have the same orbits if for almost every x ∈ X, the T 1 -orbit of x coincides with the T 2 -orbit of x. By the above remark, this is equivalent to following condition: T 1 ∈ [T 2 ] and T 2 ∈ [T 1 ]. We say that two measure-preserving transformations T 1 , T 2 ∈ Aut(X, µ) are orbit equivalent if there exists S ∈ Aut(X, µ) such that ST 1 S -1 and T 2 have the same orbits, that is

[T] := {U ∈ Aut(X, µ) : ∀x ∈ X, ∃n ∈ Z such that U(x) = T n (x)
ST 1 S -1 ∈ [T 2 ] and T 2 ∈ [ST 1 S -1 ].
Fix an aperiodic transformation T ∈ Aut(X, µ). Any U ∈ [T] is completely determined by its T-cocycle, defined as the unique function c U : X → Z satisfying the equation U(x) = T c U (x) (x) for all x ∈ X. The T-cocycle satisfies the Chapter 2 so-called cocycle identity: given U, V ∈ [T], we have

c UV (x) = c U (V(x)) + c V (x) for all x ∈ X.
(2.2) Let T ∈ Aut(X, µ) and A ⊆ X be a measurable subset. The first return time of T to A is the map n T,A : A → N * defined by n T,A (x) := min{n ∈ N * : T n (x) ∈ A}. This function is well-defined up to measure zero by Poincaré's recurrence theorem. For convenience, we extend n T,A to all X, setting it to be 0 on X \ A. Kac's lemma [START_REF] Kac | On the notion of recurrence in discrete stochastic processes[END_REF] yields the following inequality X n T,A (x)dµ ≤ 1.

(2.

3)

The first return map of T with respect to A is the transformation (iii) If P ∈ [T], then T and U have the same orbits.

T A ∈ [T] ≤ Aut(X, µ) defined by T A (x) := T n T,A (x) (x).
Proof. We first prove (i) and (ii). Clearly U D (x) = T D (x) = x for every x / ∈ D. Since D is a fundamental domain for P, for all x ∈ D and i ∈ {1, . . . , n(x) -1}, we have P i (x) / ∈ D. Since T D (x) = x for all x / ∈ D, we deduce by induction that

U i (x) = P i (x) / ∈ D for all x ∈ D and i ∈ {1, . . . , n(x) -1}.
So for all x ∈ D, one has U n(x) (x) = UU n(x)-1 (x) = T D P n(x) (x) = T D (x). This shows Item (i) and (ii). We now prove Item (iii). Clearly, U ∈ [T]. We need to show that T ∈ [U]. Observe that for almost every x, the U-orbit of x meets D: indeed, if x ∈ P i (D) for i ∈ {1, . . . , n(x) -1}, then U -i (x) = P -i (x) ∈ D. Since being in the same orbit is an equivalence relation, it is enough to show that any two points in D, which belong to the same T-orbit, are in the same U-orbit. This follows directly from (i).

We will also need the following lemma which can be proven with the same kind of arguments as above. Lemma 2.2.4. -Let U ∈ Aut(X, µ) and let A be a measurable subset of X which intersects every U-orbit. Then (U A ) -1 U is periodic and A is a fundamental domain for it.

Proof. Since A intersects every U-orbit, for almost every x ∈ X \ A there exists a smallest n ≥ 1 such that U -n (x) ∈ A. Remark that ((U A ) -1 U) -n (x) = U -n (x) ∈ A and hence A intersects every (U A ) -1 U-orbit. If x ∈ A, then for every 0 ≤ n < n U,A (x) we have that

((U A ) -1 U) n (x) = U n (x) / ∈ A and ((U A ) -1 U) n U,A (x) (x) = U -1 A UU n U,A (x)-1 (x) = x.
Since A intersects every (U A ) -1 U-orbit, we obtain both that every (U A ) -1 Uorbit is finite and that A is a fundamental domain for (U A ) -1 U.

ϕ-integrable orbit equivalence and full groups

We first define the notion of ϕ-integrable orbit equivalence.

Definition 2.2.5. -Fix ϕ : R + → R + . Two aperiodic transformations T 1 , T 2 ∈ Aut(X, µ) are ϕ-integrable orbit equivalent if there exists S ∈ Aut(X, µ) such that ST 1 S -1 and T 2 have the same orbits and their respective cocycles are ϕintegrable. To be more precise, we ask that

X ϕ(|c ST 1 S -1 (x)|)dµ < +∞ and X ϕ(|c T 2 (x)|)dµ < +∞,
where c ST 1 S -1 is the T 2 -cocycle of ST 1 S -1 and c T 2 is the ST 1 S -1 -cocycle of T 2 , defined for all x ∈ X by the equations

ST 1 S -1 (x) = T c ST 1 S -1 (x) 2 (x) and T 2 (x) = (ST 1 S -1 ) c T 2 (x) (x).
When ϕ(t) = t p for some p ∈ (0, +∞), we recover the notion of L p orbit equivalence.

Remark 2.2.6. -We warn the reader that even though the term L p orbit equivalence is often used in the literature, this terminology may sound a bit deceptive. Indeed, since the integrability condition has no reason to be preserved under composition of orbit equivalences, we do not expect ϕ-integrable (even L p ) orbit equivalence to be an equivalence relation for every concave function ϕ, although we don't have any counterexample. The fact that it is the case for p = 1 seems to be a rather artificial consequence of Belinskaya's theorem.

In our work, the function ϕ is at most linear and for our main theorems the function is assumed to be sublinear, that is lim t→+∞ ϕ(t)/t = 0. For example we are interested in the case of L p orbit equivalence for p ≤ 1, or in the case of ϕ(t) = log(1 + t).

to new1 examples of full groups, such as L p full groups [T] p for 0 < p < 1 obtained with the function ϕ(t) = t p , or else [T] log obtained with the function ϕ(t) = log(1 + t).

Remark 2.2.11. -Given a metric-compatible function ϕ, it is now straightforward to check that two aperiodic transformations T 1 , T 2 ∈ Aut(X, µ) are ϕ-integrable orbit equivalent if and only if there is S ∈ Aut(X, µ) such that ST 1 S -1 ∈ [T 2 ] ϕ and T 2 ∈ [ST 1 S -1 ] ϕ . However, the notion of ϕ-orbit equivalence is a priori weaker than conjugacy of ϕ-integrable full groups. Indeed conjugacy of ϕ-integrable full groups is an equivalence relation but ϕ-integrable orbit equivalence may not be, see Remark 2.2.6. This is in contrast with the case of classical orbit equivalence, see [Kec10, Thm. 4.1].

In our two main results, namely Theorem 2.1.2 and Theorem 2.1.3, the sublinear function ϕ is not assumed to be metric-compatible. The following lemma will allow us to reduce to the case where ϕ is in addition metric-compatible. Lemma 2.2.12. -Let ϕ : R + → R + be a sublinear function. Then there is a sublinear metric-compatible function ψ : R + → R + such that ϕ(t) ≤ ψ(t) for all t large enough.

Proof. Set

θ : R * + → R + , θ(t) := min 1, sup s≥t ϕ(s) + 1 s ; ψ : R + → R + , ψ(t) := t 0 θ(s)ds.
Noting that θ is positive-valued and non-increasing, it is straightforward to check that ψ is non-decreasing, subadditive and that ψ(t) = 0 if and only if t = 0. Moreover the fact that θ(t) tends to 0 as t approaches +∞ implies that ψ is sublinear. Now remark that for every

t ∈ R * + ψ(t) = t 0 θ(s)ds ≥ t 0 θ(t)ds = tθ(t).
For t ∈ R * + large enough so that sup s≥t ϕ(s) + 1 s ≤ 1 we finally have

tθ(t) = t sup s≥t ϕ(s) + 1 s = t sup s≥t ϕ(s) + 1 s ≥ t ϕ(t) + 1 t = ϕ(t) + 1 ≥ ϕ(t)
so we are done.

Remark 2.2.13. -Given a sublinear function ϕ, Lemma 2.2.12 grants us a sublinear metric-compatible function ψ such that ϕ(t) ≤ ψ(t) for all t large enough.

Chapter 2

Therefore, for any measurable function f : X → Z we have

X ψ(| f (x)|)dµ < +∞ implies that X φ(| f (x)|)dµ < +∞.
In particular, ψ-integrable orbit equivalence implies ϕ-orbit equivalence and any element in a ψ-integrable full group will have ϕ-integrable cocycle.

We will state most of our results in the comfortable context of (sublinear) metric-compatible functions. However, many of our statement could be easily generalized to the general context of sublinear functions through Remark 2.2.13. We will explicitly do so only in our main theorems, Theorem 2.1.2 and Theorem 2.1.3.

Metric properties of ϕ-integrable full groups

We now introduce and study a natural extended pseudo-metric on full groups from which ϕ-integrable full groups naturally arise. Lemma 2.2.14. -Let ϕ : R + → R + be a metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic transformation. Let d ϕ,T : [T] × [T] → R + ∪ {+∞} be the function defined by

d ϕ,T (U, V) := X |c U (x) -c V (x)| ϕ dµ.
Then the following are true.

(i) The group [T] ϕ is determined by d ϕ,T :

[T] ϕ = {U ∈ [T] : d ϕ,T (U, id) < +∞}. (ii) The restriction of d ϕ,T to [T] ϕ × [T] ϕ is a metric on [T] ϕ which is right-invariant, that is, for all U, V, W ∈ [T] ϕ , d ϕ,T (UW, VW) = d ϕ,T (U, V).
Proof. Item (i) is an immediate consequence of the definition of [T] ϕ .

Let us now prove Item (ii). The fact that d ϕ,T is a metric is a straightforward consequence of the fact that (s, t) → |s -t| ϕ is a metric on R. The rightinvariance follows from the cocycle identity (2.2) and the fact that the transfor-mation W is measure-preserving:

d ϕ,T (UW, VW) = X |c UW (x) -c VW (x)| ϕ dµ = X |c U (W(x)) -c V (W(x))| ϕ dµ = X |c U (x) -c V (x)| dµ = d ϕ,T (U, V).
Example 2.2.15. -Consider the metric-compatible function ϕ := min(id R + , 1). Then it is straightforward to check that d ϕ,T = d u is the uniform metric on

[T] = [T] ϕ .
Another example is obtained by taking ϕ := id R + ; we then recover the L 1 metric on the L 1 full group [T] 1 = [T] ϕ .

In order to compare ϕ-integrable full groups, we are led to compare asymptotically metric-compatible functions. We will use the following standard notation: given two real-valued functions f and g, we write f (t) = O(g(t)) as t → +∞ if there exist t 0 > 0 and C > 0 such for all t > t 0 , we have

| f (t)| ≤ C|g(t)|.
Since the functions we consider are subadditive, it is enough to compare them on the integers. Lemma 2.2.16. -Let ϕ, ψ be two metric compatible function. Then the following are equivalent.

(i) ϕ(t) = O(ψ(t)) as t → +∞.

(ii) There exists C > 0 such that ϕ(t) ≤ Cψ(t) for all t ≥ 1.

(iii) There exists C > 0 such that ϕ(k) ≤ Cψ(k) for all integer k ∈ N.

Proof. We first prove that (i) implies (ii). Let t 0 ≥ 1 and D > 0 such that for all t > t 0 , we have ϕ(t) ≤ Dψ(t). Set C := max(D, ϕ(t 0 )/ψ(1)) and observe that since ϕ and ψ are non-decreasing, ϕ(t) ≤ Cψ(t) for all t ≥ 1.

The implication (ii) =⇒ (iii) is straightforward, so we are left with proving (iii) =⇒ (i). Let C > 0 such that ϕ(k) ≤ Cψ(k) for all integer k ∈ N. Fix a real number t ≥ 2 and let n ∈ N * such that n ≤ t < n + 1. Then we have

ϕ(t) ≤ ϕ(n + 1) ≤ Cψ(n + 1) ≤ C(ψ(t) + ψ(1)) ≤ C 1 + ψ(1) ψ(t) ψ(t),
and since ψ(t) ≥ ψ(1) for every t ≥ 1, the proof is complete.

We now compare ϕ-integrable full groups for different metric-compatible functions.

Lemma 2.2.17. -Let ϕ and ψ be two metric-compatible functions and fix an aperiodic transformation T ∈ Aut(X, µ). If ϕ(t) = O(ψ(t)) as t → +∞, then [T] ψ ≤ [T] ϕ . Moreover, the inclusion map is Lipschitz.

Proof. By Lemma 2.2.16, there is

C > 0 such that ϕ(k) ≤ Cψ(k) for all integer k ∈ N. Let U, V ∈ [T]. Then for almost every x ∈ X, |c U (x) -c V (x)| ϕ ≤ C|c U (x) -c V (x)| ψ .
Integrating over X, we get that d ϕ,T (U, V) ≤ Cd ψ,T (U, V). The lemma now follows immediatly.

Corollary 2.2.18. -Whenever T is an aperiodic measure-preserving transformation, we have

[T] 1 ≤ [T] ϕ ≤ [T],
and the inclusion maps are Lipschitz, Proof. Since ϕ is subadditive, we have ϕ(t) = O(t) as t → +∞. Moreover min(1, t) = O(ϕ(t)) as t → +∞. The conclusion now follows from Lemma 2.2.17.

We will show in Proposition 2.4.2 that Lemma 2.2.17 is an equivalence. For this, we will make a crucial use of the fact that the topologies induced by these metrics are Polish group topologies, see Theorem 2.4.1.

Remark 2.2.19. -Let d T : X × X → R + ∪ {+∞} be the extended metric on X defined by d T (x, y) := inf{n ∈ N :

T n (x) = y or T n (y) = x}.
Then by definition of the T-cocycle of any U ∈ [T], we have that for all x ∈ X,

d T (U(x), x) = |c U (x)|. For all U, V ∈ [T] ϕ , the cocycle identity implies that c UV -1 (x) = c U (V -1 (x)) -c V (V -1 (x)
). Since V preserves the measure, we obtain

d ϕ,T (U, V) = X |c U (V -1 (x)) -c V (V -1 (x))| ϕ = X |c UV -1 (x)| ϕ dµ = X ϕ(d T (UV -1 (x), x))dµ = X ϕ(d T (U(x), V(x))dµ.
We won't use this formula thereafter. However, this point of view allows one to define ϕ-integrable full groups of non-necessarily free actions of finitely generated groups. Some of the arguments given in this paper work in this wider context; this will be examined in an upcoming work.

Flexibility of ϕ-integrable orbit equivalence 2.3.1 Construction of cycles in ϕ-integrable full groups

An n-cycle, n ≥ 2, is a periodic transformation P ∈ Aut(X, µ) whose orbits have cardinality either 1 or n. The aim of this section is to prove the following result.

Theorem 2.3.1. -Let ϕ : R + → R + be a sublinear metric-compatible function. Let T ∈ Aut(X, µ) be aperiodic. Then for all measurable A ⊆ X and all integer n ≥ 2, there exists an n-cycle P ∈ [T] ϕ whose support is equal to A.

Remark ⊗Z , where κ is the uniform measure on {0, 1}. Then for every 0 < p < 1/2, there exists an involution in [T] p with full support and fundamental domain X 0 := {(x n ) n∈Z ∈ {0, 1} Z : x 0 = 0}. Indeed, for all x ∈ X 0 , let N(x) be the infimum of n ≥ 1 such that 1 appears strictly more often than 0 in {x 1 , . . . , x n }. Then the map π : x ∈ X 0 → T N(x) (x) ∈ {0, 1} Z \ X 0 is almost everywhere well-defined and injective. Thus it can be extended to an involution P ∈ [T] with full support and fundamental domain X 0 . Standard estimates on the simple random walk on Z imply that P belongs to [T] p for all 0 < p < 1/2. Remark 2.3.4. -Theorem 2.3.1 tells us that any measurable subset A ⊆ X is the support of an involution. The situation is less flexible regarding fundamental domains. For example, the subset X 0 introduced in the previous example cannot be the fundamental domain of any involution in the L p full group of the Bernoulli shift for 1/2 ≤ p ≤ 1, as a consequence of a result of Liggett [Lig02]. Note that his result is more general and stated in probabilistic terms; the connection to our context and a purely ergodic-theoretic version of his proof will be presented in the second named author's PhD thesis.

A partial measure-preserving transformation of (X, µ) is a bimeasurable measure-preserving bijection π between two measurable subsets dom(π) and rng(π) of X, called respectively the domain and the range of π. The support of π is the set

supp(π) := {x ∈ dom(π) : π(x) = x} ∪ {x ∈ rng(π) : π -1 (x) = x}.
A pre-cycle of length n ≥ 2 is a partial measure-preserving transformation π : dom(π) → rng(π) of (X, µ) such that if we set B := dom(π) \ rng(π), then

• {π 0 (B), . . . , π n-2 (B)} is a partition of dom(π),

• {π 1 (B), . . . , π n-1 (B)} is a partition of rng(π).

The set B = dom(π) \ rng(π) is called the basis of the pre-cycle π.

A pre-cycle π of length n can be extended to an n-cycle P and called the closing cycle of π, as follows:

P(x) :=      π(x) if x ∈ dom(π), π -(n-1) (x) if x ∈ rng(π) \ dom(π), x else.
Observe that the support of P coincides with the support of the pre-cycle π and that the basis B is a fundamental domain for the restriction of P to its support. A pre-cycle π is induced by T ∈ Aut(X, µ) if for all x ∈ dom(π), we have π(x) = T supp(π) (x).

Lemma 2.3.5. -Let ϕ : R + → R + be a metric-compatible function. Let T ∈ Aut(X, µ) be an aperiodic transformation, let π be a pre-cycle induced by T and let P be its closing cycle. Then d ϕ,T (P, id) ≤ 2d ϕ,T (T supp(π) , id).

In particular P belongs to [T] ϕ .

Proof. Let n be the length of the pre-cycle π, let A := supp(π) and let B := dom(π) \ rng(π) the basis of π. Since π is induced by T, for all x ∈ dom(π), one has π(x) = P(x) = T A (x). This implies that c P (x) = c T A (x) for all x ∈ dom(π). Thus,

d ϕ,T (P, id) = dom(π) |c T A (x)| ϕ dµ + P n-1 (B) |c P -(n-1) (x)| ϕ dµ ≤ d ϕ,T (T A , id) + B |c P n-1 (x)| ϕ dµ.
Moreover, for all x ∈ B, the cocycle identity yields

|c P n-1 (x)| ϕ ≤ |c P (x)| ϕ + |c P (P(x))| ϕ + . . . |c P (P n-2 (x))| ϕ .
We now use the fact that P preserves the measure and that dom(π) = B P(B)

• • • P n-2 (B) to get B |c P n-1 (x)| ϕ dµ ≤ dom(π) |c P (x)| ϕ dµ ≤ X |c T A (x)| ϕ dµ,
which concludes the proof.

Kac's lemma, that is Equation (2.3), implies that for every measurable A ⊆ X, the first return map T A belongs to [T] 1 , which is contained in [T] ϕ for every metric-compatible function ϕ by Corollary 2.2.18. We will need a more quantitative version of this fact. Lemma 2.3.6. -Let ϕ : R + → R + be a metric-compatible function. Let T ∈ Aut(X, µ) be an aperiodic transformation, let A ⊆ X be a measurable subset and let C > 0. Then

d ϕ,T (T A , id) ≤ Cϕ(1)µ(A) + sup t>C ϕ(t) t .
Proof. Recall that the T-cocycle of T A is the return time n T,A , which is nonnegative. Set B := {x ∈ A : n T,A (x) ≤ C}. We have

d ϕ,T (T A , id) = B ϕ(n T,A (x))dµ + A\B ϕ(n T,A (x))dµ ≤ Cϕ(1)µ(B) + A\B ϕ(n T,A (x)) n T,A n T,A (x)dµ ≤ Cϕ(1)µ(A) + sup t>C ϕ(t) t A\B n T,A (x)dµ
and the last integral is at most 1 by Kac's lemma, see Equation (2.3).

Corollary 2.3.7. -Let ϕ : R + → R + be a sublinear metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic transformation. Then d ϕ,T (T A , id) tends to 0 as µ(A) approaches 0.

Proof. Fix ε > 0. By sublinearity, let C > 0 such that for all t > C, we have ϕ(t)/t < ε. For all measurable A ⊆ X, if µ(A) < ε/Cϕ(1), then d ϕ,T (T A , id) < 2ε, which concludes the proof.

Remark 2.3.8. -In particular, by taking ϕ bounded, we recover the well-known fact that d u (T A , id) tends to 0 as µ(A) approaches 0 (see Lemma 2.4.9).

The following lemma is a direct consequence of Rokhlin's lemma.

Lemma 2.3.9. -Let T ∈ Aut(X, µ) be aperiodic and A ⊆ X be measurable. For all ε > 0 and all integer n ≥ 2, there exists a pre-cycle π of length n, induced by T, such that supp(π) ⊆ A and µ(A \ supp(π)) ≤ ε.

Proof. Since T is aperiodic, T A is aperiodic on its support. We apply Rokhlin's lemma to T A to find a measurable subset B ⊆ A such that B, T A (B), . . . , (T A ) n-1 (B) are pairwise disjoint and

µ A \ B • • • (T A ) n-1 (B) ≤ ε.
Then the restriction of

T A to B • • • (T A ) n-2 (B) is a pre-cycle of length n,
which is induced by T A and thus by T. Finally, its support satisfies the desired assumptions.

We are now ready to prove the existence of n-cycles with prescribed support in ϕ-integrable full groups.

Proof of Theorem 2.3.1. Let T ∈ Aut(X, µ) be an aperiodic element, let A ⊆ X be a measurable subset and let n ≥ 2. Since ϕ is sublinear, we can and do fix a sequence (C k ) k≥1 of strictly positive numbers such that sup

t>C k ϕ(t) t ≤ 2 -k for all k ≥ 1.
Then, we use Lemma 2.3.9 to construct inductively a sequence (π k ) k≥0 of precycles of length n induced by T, whose supports are pairwise disjoint subsets of A and such that for all k ≥ 1,

µ A \ supp(π 0 ) • • • supp(π k-1 ) ≤ 1 2 k C k .
This inequality implies in particular that for all k ≥ 1, we have µ(supp(π k )) ≤ 1/(2 k C k ). Let P k be the closing cycle of π k and let P ∈ Aut(X, µ) be the n-cycle defined by P(x) := P k (x) for x ∈ supp(P k ) and P(x) := x for x / ∈ A. The support of P is equal to A and by Lemma 2.3.5 and Lemma 2.3.6, we have

d ϕ,T (P, id) = ∑ k≥0 d ϕ,T (P k , id) ≤ 2 ∑ k≥0 d ϕ,T (T supp(π k ) , id) ≤ 2d ϕ,T (T supp(π 0 ) , id) + 2 ∑ k≥1 ϕ(1)C k µ(supp(π k )) + sup t>C k ϕ(t) t .
The second term is by construction a converging series, so we are done.

Construction of ϕ-integrable orbit equivalences

Let us now prove Theorem 2.1.3.

Theorem 2.3.10. -Let ϕ : R + → R + be a sublinear function. Let T ∈ Aut(X, µ) be ergodic. For all n ≥ 2, there exists U ∈ Aut(X, µ) such that T and U are ϕ-integrable orbit equivalent and U n is not ergodic.

Proof. By Lemma 2.2.12, there is a sublinear metric-compatible function ψ such that ϕ(t) ≤ ψ(t) for all t large enough. In particular, ψ-integrable orbit equivalence implies ϕ-orbit equivalence (cf. Remark 2.2.13). Hence if the theorem holds for ψ then it holds for ϕ. Therefore, by replacing ϕ by ψ, we may and do assume that ϕ is a metric-compatible function. By Theorem 2.3.1, there exists an n-cycle P ∈ [T] ϕ whose support is X. We fix a fundamental domain D for P and we let U := T D P. By Lemma 2.2.3 the following hold:

• the first return maps U D and T D coincide: U D = T D ;

• for all x ∈ D, we have U D (x) = U n (x);

• T and U have same orbits.

By the second item, the set D is U n -invariant. So U n is not ergodic.

We will now prove that T and U are ϕ-integrable orbit equivalent. Since T and U have same orbits, we are left to show that T ∈ [U] ϕ and U ∈ [T] ϕ . As a direct consequence of Kac's lemma, see Equation (2.3), we have that

T D ∈ [T] 1 ≤ [T] ϕ and therefore U = T D P ∈ [T] ϕ .
We now prove that T ∈ [U] ϕ . In the sequel, if a measure-preserving transformation V belongs to [T] = [U], we shall denote by c T V the T-cocycle of V and by

c U V the U-cocycle of V. Claim. -Let V ∈ [T]. Then for all y ∈ D such that V(y) ∈ D, c U V (y) ≤ n c T V (y) .
Proof of Claim. Note that since y and V(y) belong to D, any i ∈ Z such that U i (z) = V(z) must be a multiple of n. If we combine this with the fact that U D (z) = U n (z) for all z ∈ D and that U D = T D , we obtain:

c U V (y) = min{|i| : U i (y) = V(y)} =n min{|i| : U i D (y) = V(y)} =n min{|i| : T i D (y) = V(y)} ≤n min{|i| : T i (y) = V(y)} ≤n c T V (y) . Let x ∈ X. By definition of U, there are two integers 0 ≤ k, l ≤ n -1 such that U k (x) ∈ D and U l (T(x)) ∈ D. By the cocycle identity, c U U l TU -k (U k (x)) = c U U l (T(x)) + c U T (x) + c U U -k (U k (x)) = l + c U T (x) -k. Hence c U T (x) ≤ c U U l TU -k (U k (x)) + n.
Using the claim for V = U l TU -k and y = U k (x), we obtain

c U T (x) ≤ n c T U l TU -k (U k (x)) + n.
Integrating over X, we get

X c U T (x) ϕ dµ ≤ max 0≤k,l≤n-1 X n c T U l TU -k (U k (x)) ϕ dµ + ϕ(n),
which is bounded since U l TU -k ∈ [T] ϕ . Hence T ∈ [U] ϕ and this concludes the proof of the theorem.

The following direct corollary says that the analogue of Belinskaya's theorem for ϕ-integrable orbit equivalence does not hold as soon as ϕ is sublinear.

Corollary 2.3.11. -Let ϕ : R + → R + be a sublinear function. Let T ∈ Aut(X, µ) be an ergodic transformation and assume that T n is ergodic for some n ≥ 2. Then there exists U ∈ Aut(X, µ) such that T and U are ϕ-integrable orbit equivalent but not flip-conjugate.

Proof. Let n ≥ 2 such that T n is ergodic. By the previous theorem, we find U ∈ Aut(X, µ) such that U is ϕ-integrable orbit equivalent to T and U n is not ergodic. In particular U cannot be flip-conjugate to T because otherwise U n would be flip-conjugate to T n which is ergodic. Question 2.3.12. -Let ϕ : R + → R + be a sublinear metric-compatible function. Let T ∈ Aut(X, µ) be an ergodic transformation such that T n is non-ergodic for all n ≥ 2. Does there exist U ∈ Aut(X, µ) such that T and U are ϕ-integrable orbit equivalent but not flip-conjugate? As we will see in Section 2.4.5, a weaker result holds in full generality: for every ergodic T ∈ Aut(X, µ) and every sublinear metric-compatible function, there is U ∈ [T] ϕ such that U and T have the same orbits, but are not flipconjugate. This relies on the Baire category theorem, using the fact that [T] ϕ is a Polish group (see Section 2.4.1).

Connection to Shannon orbit equivalence

Let I be a countable set and f : X → I a measurable map. The Shannon entropy of f is the quantity

H( f ) := -∑ i∈I µ( f -1 {i}) log µ( f -1 {i}).
Definition 2.3.13 (Kerr-Li). -Two aperiodic transformations T 1 , T 2 ∈ Aut(X, µ) are Shannon orbit equivalent if there exists S ∈ Aut(X, µ) such that ST 1 S -1 and T 2 have the same orbits and

H(c ST 1 S -1 ) < +∞ and H(c T 2 ) < +∞, where c ST 1 S -1 is the T 2 -cocycle of ST 1 S -1 and c T 2 is the ST 1 S -1 -cocycle of T 2 .
Lemma 2.3.14. -There are two positive constants C 1 , C 2 > 0 such that for any measurable function f : X → Z, we have

H( f ) ≤ C 1 X log(1 + | f (x)|)dµ + C 2 .
The proof we propose is inspired by a classical proof that integrable functions have finite Shannon entropy, see for instance [Aus16, Lem. 2.1] or [Dow11, Fact 1.1.4].

Proof. Let f + := max( f , 0) and f -:= min( f , 0), so that f = f + + f -. We have

X log(1 + | f |)dµ = X log(1 + f + )dµ + X log(1 -f -)dµ.
By subadditivity, see for instance [Dow11, Chap. 1],

H( f ) = H( f + + f -) ≤ H( f + ) + H(-f -).
Hence, it is enough to prove the lemma for f : X → N. So let us fix f : X → N. For all n ∈ N, let p n := µ( f -1 {n}). By definition of the Shannon entropy,

H( f ) = -∑ n≥0 p n log p n .
Studying the variations of the function g t (s) = st + e -s-1 , one checks that for all t > 0 and s ∈ R, -t log t ≤ st + e -s-1 . Applying this for t = p n and s = 2 log(n + 1) and summing over n, we get

H( f ) ≤ 2 ∑ n≥0 p n log(1 + n) + ∑ n≥0 e -1 (n + 1) 2 .
To conclude, we observe that ∑ n≥0 p n log(1 + n) = X log(1 + f (x))dµ.

We immediately deduce the following comparison between ϕ-integrable orbit equivalence and Shannon orbit equivalence. Theorem 2.3.15. -Let ϕ : R + → R + be a function such that log(1 + t) = O(ϕ(t)) as t → +∞. Then for any aperiodic transformation T ∈ Aut(X, µ), every S ∈ [T] whose T-cocycle is ϕ-integrable has finite Shannon entropy.

In particular, if two aperiodic transformations S, T ∈ Aut(X, µ) are ϕ-integrable orbit equivalent, then they are Shannon orbit equivalent.

Remark 2.3.16. -Note that for every p ∈ (0, +∞), we have log(1 + t) = O(t p ) as t → +∞. Therefore L p orbit equivalence implies Shannon orbit equivalence for measure-preserving transformations.

In [START_REF] Kerr | Entropy, Shannon orbit equivalence, and sparse connectivity[END_REF], Kerr and Li asked whether Shannon orbit equivalence of ergodic transformations implies flip-conjugacy. We prove that it is not the case.

Theorem 2.3.17. -Let T ∈ Aut(X, µ) be an ergodic transformation, assume that T n is ergodic for some n ≥ 2. Then there exists U ∈ Aut(X, µ) such that T and U are Shannon orbit equivalent but not flip-conjugate.

Proof. Let us consider the sublinear metric-compatible function ϕ : R + → R + given by ϕ(t) := log(1 + t). By Corollary 2.3.11, there exists U ∈ Aut(X, µ) such that T and U are ϕ-integrable orbit equivalent, but not flip-conjugate. On the other hand, the transformations T and U are Shannon orbit equivalent by Theorem 2.3.15.

Finiteness of entropy and Shannon orbit equivalence

Kerr and Li implicitly asked whether dynamical entropy is an invariant of Shannon orbit equivalence for ergodic measure-preserving transformations. Shortly after a first version of our paper appeared, they obtained a positive answer [KL22, Thm. A]. In this section, we provide a short proof that finiteness of dynamical entropy is an invariant of Shannon orbit equivalence. We start by recalling a definition of dynamical entropy of measure-preserving transformations which is convenient for our purposes. Definition 2.3.18. -Let T ∈ Aut(X, µ). A measurable map f : X → I, where I is countable, is called T-dynamically generating if there is a full measure set X 0 ⊆ X such that for all distinct x, y ∈ X 0 , there is n ∈ Z such that f (T n (x)) = f (T n (y)). Proof. Let f : X → I be a U-dynamically generating function and denote by c U the T-cocycle of U. We claim that the couple

( f , c U ) : X → I × Z is T- dynamically generating. Indeed, let x, y ∈ X such that c U (T n (x)) = c U (T n (y)) and f (T n (x)) = f (T n (y)) for all n ∈ Z.
The first equality and the cocycle identity imply that c

U n (x) = c U n (y) for all n ∈ Z. So for all n ∈ Z f (U n (x)) = f (T c U n (x) (x)) = f (T c U n (y) (y)) = f (U n (y)).
Since f is U-dynamically generating, the above equation implies that x = y.

Assume by contradiction that U has finite entropy. This implies that there exists a U-generating function f with finite Shannon entropy. Since c U has finite Shannon entropy, then so does the function ( f , c U ). But we have seen that ( f , c U ) is a T-generating function, hence we deduce that T has finite dynamical entropy and the proof is complete.

Corollary 2.3.21. -Suppose T 1 , T 2 are two aperiodic measure-preserving transformations which are Shannon orbit equivalent. Then T 1 has finite dynamical entropy if and only if T 2 has finite dynamical entropy.

Remark 2.3.22. -As explained before, Kerr and Li recently obtained that dynamical entropy itself is preserved under Shannon orbit equivalence [KL22, Thm. A]. Suppose now that ϕ : R + → R + is a function that satisfies log(1 + t) = O(ϕ(t)) as t → +∞, such as ϕ(t) = log(1 + t) or ϕ(t) = t p . By Kerr and Li's result and Theorem 2.3.15, dynamical entropy is an invariant of ϕ-integrable orbit equivalence. In particular it is an invariant of L p orbit equivalence for p ∈ (0, +∞). When ϕ is moreover sublinear, this is the only invariant of ϕintegrable orbit equivalence that we know for ergodic transformations, even for L p orbit equivalence where p ∈ (0, 1).

Weakly mixing elements are generic in [T] ϕ

This last section is dedicated to the proof of Theorem 2.1.2: we are going to show that for every sublinear metric-compatible function ϕ and ergodic transformation T, there is an element U ∈ [T] ϕ which has the same orbit as T but is not flip-conjugated to T.

Note that we have shown in Corollary 2.3.11 that this is already the case if T is an ergodic transformation such that T n is ergodic for some n ≥ 2. Therefore we will restrict ourselves to the case when there exists n ≥ 2 such that T n is not ergodic. For such transformations, we will not construct any explicit U ∈ [T] ϕ , but we will use the Baire category theorem. We will show that given an aperiodic transformation T, the possible candidates of such U are generic, see Theorem 2.4.15.

We start with three preparatory sections to introduce the required material. We believe them to be of independent interest. In the first one, we show that the metric d ϕ,T is a complete separable metric inducing a Polish group topology on [T] ϕ , see Theorem 2.4.1. In the second one, we prove a sublinear ergodic theorem in the context of ϕ-integrability which will play a crucial role later on. In the third one, we study continuity properties of the first return map.

Polish group topology

Recall that the full group [T] ≤ Aut(X, µ) is closed and separable for the topology induced by the uniform metric d u and therefore it is a Polish group [Kec10, Prop. 3.2]. We shall see that ϕ-integrable full groups provide further interesting classes of Polish groups.

Let ϕ : R + → R + be a metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic transformation. We introduced the ϕ-integrable full group [T] ϕ as the group of measure-preserving transformation whose cocycle is ϕ-integrable. In Lemma 2.2.14, we defined a metric d ϕ,T on [T] ϕ . The goal of this section is to prove that the topology induced by d ϕ,T on [T] ϕ is a Polish group topology.

U ∈ [T] be its d u -limit. Then U ∈ [T] ϕ and lim n d ϕ,T (U n , U) = 0. In particular d ϕ,T is complete. Proof of the claim. Since (U n ) n≥0 is d ϕ,T -Cauchy, there is m such that for all n ≥ m, X |c U n (x)| ϕ dµ = d ϕ,T (U n , id) ≤ d ϕ,T (U n , U m ) + d ϕ,T (U m , id) ≤ 1 + d ϕ,T (U m , id).
Moreover since lim n d u (U n , U) = 0, we have that (c U n ) n≥0 converges in measure to c U and thus a subsequence of (c U n ) n≥0 converges pointwise to c U . Fatou's lemma then implies that U ∈ [T] ϕ . The triangle inequality for |

•| ϕ gives X |c U n (x) -c U (x)| ϕ -|c U m (x) -c U (x)| ϕ dµ ≤ X |c U n (x) -c U m (x)| ϕ dµ = d ϕ,T (U n , U m ), hence the sequence (|c U n -c U | ϕ ) n≥0 is Cauchy with respect to the L 1 -metric. Since (c U n -c U ) n≥0 converges in measure to 0, we must have that lim n d ϕ,T (U n , U) = lim n X |c U n (x) -c U (x)| ϕ dµ = 0 so the claim is proved. claim
Let us now show that the topology induced by d ϕ,T is a group topology. We start by proving the continuity of the inverse map. Let (U n ) n≥0 be a sequence of elements of [T] ϕ converging to U ∈ [T] ϕ . Then the cocycle identity gives us that

0 = c UU -1 (x) = c U (U -1 (x)) + c U -1 (x) and hence d ϕ,T (U -1 n , U -1 ) = X |c U -1 n (x) -c U -1 (x)| ϕ dµ = X |c U n (U -1 n (x)) -c U (U -1 (x))| ϕ dµ = X c U n (x) -c U (U -1 U n (x)) ϕ dµ ≤ X |c U n (x) -c U (x)| ϕ dµ + X |c U (x) -c U (U -1 U n (x))| ϕ dµ.
Since (U n ) n≥0 converges to U for the metric d ϕ,T and thus for the uniform metric, the right hand side converges to 0 and hence the inverse map is continuous. We now prove that the multiplication map is continuous. Let (U n ) n≥0 and (V m ) m≥0 be two sequences which d ϕ,T -converge to U and V respectively. Then by the triangle inequality and right-invariance,

d ϕ,T (U n V n , UV) ≤ d ϕ,T (U n , U) + d ϕ,T (UV n , UV).

Now remark that since the inverse map is continuous, UV n converges to UV if and only if

V -1 n U -1 converges to V -1 U -1
. By right-invariance and continuity of the inverse lim n d ϕ,T (V -1 n U -1 , V -1 U -1 ) = 0, which finishes the proof that d ϕ,T induces a group topology on [T] ϕ .

We are left to show that this topology is separable. Consider the following abelian group where we identify functions up to a null set [T] ϕ is isometric to a metric subspace of L ϕ (X, Z). We now prove that L ϕ (X, Z) is separable: identify X with [0, 1] equipped with the Lesbegue measure and observe that the subgroup generated by characteristic functions of rational intervals is dense. Since subspaces of separable metric spaces are separable, we conclude that [T] ϕ is separable.

L ϕ (X, Z) := f : X → Z : X | f (x)| ϕ dµ < +∞ , endowed with the metric ( f , g) → X | f (x) -g(x)| ϕ dµ. The function which takes U ∈ [T] ϕ to c U ∈ L ϕ (X, Z) is an isometry. So
We now exploit the Polish group topology to characterize the inclusion between ϕ-integrable full groups in terms of metric comparisons. In particular [T] ϕ = [T] ψ as soon as ϕ and ψ are not bi-Lipschitz. However we do not know how to construct any explicit element in [T] ϕ \ [T] ψ . Proposition 2.4.2. -Let ϕ and ψ be two metric-compatible functions and let T ∈ Aut(X, µ) be an aperiodic measure-preserving transformation. Then the following are equivalent:

(i) ϕ(t) = O(ψ(t)) as t → +∞. (ii) [T] ψ ≤ [T] ϕ .
The proof uses the following well-known lemma.

Lemma 2.4.3. -Let G be a Polish group, let H 1 ≤ H 2 ≤ G be two subgroups of G. Suppose that H 1 and H 2 are endowed with a Polish topology which refines the topology induced by G. Then the topology of H 1 refines the topology induced by H 2 .

Proof. By hypothesis, the inclusions H 1 → G and H 2 → G are continuous. In particular, the Borel structure induced by each of their topology refines the Borel structure induced by the one of G. The Lusin-Souslin theorem states that given any two Polish spaces X and Y, if f : X → Y is Borel and injective then for every Borel A ⊆ X, the set f (A) is Borel, see [Kec95, Thm. 15.1]. Therefore, we can apply it to the inclusions H 1 → G and H 2 → G to obtain that the Borel structures induced by the respective topologies of H 1 and H 2 coincide with the σ-algebra induced by the Borel subsets of G. This in particular tells us that the inclusion map H 1 → H 2 is Borel, so it is automatically continuous by Pettis' lemma [Kec95, Thm. 9.9] which proves the lemma.

Proof of Proposition 2.4.2. The implication (i)⇒(ii) follows from Lemma 2.2.17, so we only need to prove (ii)⇒(i). We argue by contradiction: assume that [T] ψ ≤ [T] ϕ but (i) does not hold. By Lemma 2.2.16, there exists a sequence (k n ) n≥0 of positive integers such that We will obtain our contradiction by constructing a sequence (U n ) n≥0 of elements of [T] ψ such that

lim n→+∞ ψ(k n ) ϕ(k n ) = 0. ( 2
d ϕ,T (U n , id) → 0 but d ψ,T (U n , id) → 0.
By Rokhlin's lemma, one can find for every n ∈ N, a measurable subset

A n ⊆ X such that A n , T(A n ), . . . T 2k n -1 (A n ) are pairwise disjoint and µ(A n ) ≥ 1 4k n . Note that µ k n -1 i=0 T i (A n ) ≥ 1 4 .
Hence, for all n such that ϕ(k n ) ≥ 4, we can pick a measurable subset B n ⊆

k n -1 i=0 T i (A n ) of measure exactly 1 ϕ(k n ) . We then define U n ∈ [T] ϕ by U n (x) :=    T k n (x) if x ∈ B n ; T -k n (x) if x ∈ T k n (B n ); x otherwise. By construction d ϕ (U n , id) = 2µ(B n )ϕ(k n ) = 1 2 but Equation (2.5) implies that d ψ (U n , id) = 2µ(B n )ψ(k n ) → 0, a contradiction.
Corollary 2.4.4. -Let ϕ and ψ be two metric-compatible functions, let T ∈ Aut(X, µ) be aperiodic. Then [T] ϕ = [T] ψ if and only if ϕ(t) = O(ψ(t)) and ψ(t) = O(ϕ(t)) as t → +∞.

A sublinear ergodic theorem for ϕ-integrable functions

In this section, we prove the following sublinear ergodic theorem, which will be a key tool in our analysis of the first return map. Given a measurable function ϕ : R + → R + , a measurable function f :

X → C is ϕ-integrable when X ϕ(| f (x)|)dµ < +∞. Theorem 2.4.5. -Let ϕ : R + → R + be a sublinear non-decreasing function. Let U ∈ Aut(X, µ) and f : X → C a measurable function f which is ϕ-integrable. Then for almost every x ∈ X lim n 1 n ϕ n-1 ∑ k=0 f (U k (x)) = 0.
The convergence also holds in L 1 , that is

lim n X 1 n ϕ n-1 ∑ k=0 f (U k (x)) dµ = 0.
Proof. We start by restricting ourselves to the subadditive case. For this, we first use Lemma 2.2.12 to find a sublinear metric-compatible function ψ and C > 0 such that ϕ(t) ≤ ψ(t) for all t ≥ C. We define φ(t) := ψ(t) + ϕ(C), then φ is still sublinear, non-decreasing and subadditive. Moreover, since ψ is non-decreasing we now have ψ(t) ≤ φ(t) for all t ≥ 0. Up to replacing ϕ by φ, we thus may and do assume that ϕ is subadditive.

Given n ≥ 1 and a ϕ-integrable function f , let for all x ∈ X g n (x

) := 1 n ϕ n-1 ∑ k=0 f (U k (x)) .
Using the fact that ϕ is subadditive, we deduce that the sequence of functions (g n ) n≥1 satisfies Kingman's subadditivity property: for all n, m ≥ 1 and all x ∈ X, g n+m (x) ≤ g n (x) + g m (U n (x)).

Kingman's subadditive theorem [START_REF] John | The ergodic theory of subadditive stochastic processes[END_REF] implies that (g n ) n≥0 converges almost everywhere to some function g and our aim becomes to show that g = 0. Recall that a sequence that converges in L 1 admits an almost surely converging subsequence. In order to prove that g = 0, it is therefore enough to prove that g n 1 converges to 0, namely to establish the second part of the theorem. To this end, let f be a ϕ-integrable function and let ε > 0. Since ϕ(| f |) is integrable and ϕ is non-decreasing, we find a measurable subset A ⊆ X and

K ≥ 0 such that X\A ϕ(| f (x)|)dµ ≤ ε and | f (x)| ≤ K for every x ∈ A.
For every measurable subset B ⊆ X, we denote f B := f 1 B , where 1 B is the indicator function of B. With this notation at hand, using first that ϕ is subadditive nondecreasing and then that U preserves the measure we obtain:

lim sup n X 1 n ϕ n-1 ∑ k=0 f X\A (U k (x)) dµ ≤ lim sup n X 1 n n-1 ∑ k=0 ϕ | f X\A (U k (x))| dµ = X ϕ | f X\A (x)| dµ ≤ ε.
Besides, since f A is bounded by K, we have for all x ∈ X

1 n ϕ n-1 ∑ k=0 f A (U k (x)) ≤ ϕ(nK) n = K ϕ(nK) nK .
Integrating over X, we obtain

X 1 n ϕ n-1 ∑ k=0 f A (U k (x)) dµ ≤ K ϕ(nK) nK .
Using that f = f X\A + f A and subadditivity, we deduce lim sup

n X 1 n ϕ n-1 ∑ k=0 f (U k (x)) dµ ≤ ε + lim sup n K ϕ(nK) nK
Since ϕ is sublinear, we finally obtain

lim sup n X 1 n ϕ n-1 ∑ k=0 f (U k (x)) dµ ≤ ε.
This proves that g n 1 converges to 0, thus ending the proof of the theorem.

Here is our main application, which will be a key tool in the following section.

Corollary 2.4.6. -Let ϕ be a sublinear metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic transformation. Then for every U ∈

[T] ϕ , lim n d ϕ,T (U n , id) n = 0.
Proof. For all integer n ≥ 0 and all x ∈ X, by the cocycle identity and the triangular inequality we have

|c U n (x)| ≤ n-1 ∑ k=0 |c U (U k (x))|.
We apply Theorem 2.4.5 to the function f (x) := |c U (x)| and get that

d ϕ,T (U n , id) n ≤ X 1 n n-1 ∑ k=0 f (U k (x)) ϕ -→ n→+∞ 0.
Remark 2.4.7. -We do not fully understand the asymptotic of the sequence (d ϕ,T (U n , id)) n≥0 . For instance, when does the sequence (d ϕ,T (U n , id)/ϕ(n)) n≥0 converge?

Continuity properties of the first return map

In the coming section we are primarily interested in continuity properties of the first return map. An important preliminary step is the following analogue of Kac's Lemma, saying that ϕ-integrable full groups are stable under first return maps.

Lemma 2.4.8. -Let T ∈ Aut(X, µ) be an aperiodic transformation and let ϕ : R + → R + be a metric-compatible function. For all U ∈ [T] ϕ and all measurable subset A ⊆ X, we have

d ϕ,T (U A , id) ≤ d ϕ,T (U, id). In particular, U A ∈ [T] ϕ .
Proof. For every integer j ≥ 1, set A j := {x ∈ A : n U,A (x) = j} where n U,A is the first return time of U to A as defined in Section 4.1. Then,

X |c U A (x)| ϕ dµ = A c U A (x) ϕ dµ = +∞ ∑ j=1 A j |c U A (x)| ϕ dµ = +∞ ∑ j=1 A j |c U j (x)| ϕ dµ
By the cocycle identity, for every j ≥ 1 we have c U j (x) = ∑ j-1 i=0 c U (U i (x)), so by the triangle inequality we obtain

X |c U A (x)| ϕ dµ ≤ +∞ ∑ j=1 j-1 ∑ i=0 A j |c U (U i (x))| ϕ dµ ≤ +∞ ∑ j=1 j-1 ∑ i=0 U i (A j ) |c U (x)| ϕ dµ ≤ X |c U (x)| ϕ dµ,
the last inequality being a consequence of the fact that the sets U i (A j ) are disjoint for j ∈ N and i ∈ {0, . . . , j -1}.

In order to state the continuity properties of the first return map, let us first observe that since we are working up to measure zero, the first return map with respect to a set A only depends on A up to a null set. It is therefore natural to introduce the measure algebra MAlg(X, µ), defined as the algebra of measurable subsets modulo identifying subsets which differ on a null set. We endow MAlg(X, µ) with the metric d µ (A, B) := µ(A B). and we can therefore conclude the proof

X | f -g| ϕ dµ = A | f -g| ϕ dµ ≤ A | f | ϕ dµ + A |g| ϕ dµ ≤ 3ε.
We can now prove the following proposition which is the ϕ-integrable analogue of [LM18, Prop. 3.9]. Proposition 2.4.12. -Let ϕ be a metric-compatible function and T ∈ Aut(X, µ) an ergodic transformation. Consider U ∈ [T] ϕ and consider a measurable subset A ⊆

X. If (A n ) n≥0 is a sequence of measurable subsets of A such that lim n µ(A \ A n ) = 0, then lim n d ϕ,T (U A n , U A ) = 0.
Proof. Since lim n µ(A \ A n ) = 0 and since the first return map is continuous with respect to the uniform metric by Lemma 2.4.9, we get that lim n d u (U A n , U A ) = 0. This means that (c U A n ) n≥0 converges in measure to c U A and therefore that (|c

U A n | ϕ ) n≥0 converges in measure to |c U A | ϕ . Thanks to Lemma 2.4.8, we have for all n ≥ 0, d ϕ,T (U A n , id) ≤ d ϕ,T (U A , id). In other words X |c U A n (x)| ϕ dµ ≤ X |c U A (x)| ϕ dµ.
Hence we can apply Lemma 2.4.11, yielding lim

n X |c U A n (x) -c U A (x)| ϕ dµ = 0.
This precisely means that lim n d ϕ,T (U A n , U A ) = 0, so we are done.

Let ϕ : R + → R + be a sublinear metric-compatible function and T ∈ Aut(X, µ) be an aperiodic transformation. In Corollary 2.3.7, we proved that for any aperiodic transformation T ∈ Aut(X, µ), the quantity d ϕ,T (T A , id) tends to 0 as µ(A) approaches 0. It is natural to ask whether this holds for all aperiodic U ∈ [T] ϕ , i.e. does d ϕ,T (U A , id) tends to 0 as µ(A) approaches 0? We were not able to answer this question, but we can prove the following much weaker statement. Its proof relies on our sublinear ergodic theorem (Theorem 2.4.5), or rather on Corollary 2.4.6. Proposition 2.4.13. -Let ϕ be a sublinear metric-compatible function. Let T ∈ Aut(X, µ) be an aperiodic transformation. Then for any aperiodic transformation U ∈ [T] ϕ and for any measurable subset A ⊆ X, there exists a sequence (A k ) k≥0 of measurable subsets contained in A which intersect every U-orbit, such that lim k µ(A k ) = 0 and lim k d ϕ,T (U A k , id) = 0.

Proof. Put V := U A and remark that for every measurable B ⊆ A, we have that V B = U B . As an immediate consequence of Alpern's multiple Rokhlin theorem [START_REF] Alpern | Generic properties of measure preserving homeomorphisms[END_REF] 2 , for every k ≥ 0, one can find a measurable subset B k ⊆ A which meets every V-orbit in A and such that n V,B k (B k ) = {k, k + 1}. The latter implies that the V i (B k ) are disjoint for i ∈ {0, ..., k -1}. Observe that for all x ∈ X and i ∈ Z we have n

V,V i (B k ) (x) = n V,B k (V -i (x)). This implies that for all x ∈ V i (B k ), either V V i (B k ) (x) = V k (x) or V V i (B k ) (x) = V k+1 (x). Therefore by integrating over the disjoint union of the V i (B k ) for i ∈ {0, ..., k -1} we get that k-1 ∑ i=0 d ϕ,T (V V i (B k ) , id) ≤ d ϕ,T (V k , id) + d ϕ,T (V k+1 , id), whence there exists 0 ≤ i k ≤ k -1 such that d ϕ,T (V V i k (B k ) , id) ≤ d ϕ,T (V k , id) + d ϕ,T (V k+1 , id) k .
The set A k := V i k (B k ) has measure less than 1/k. Corollary 2.4.6 implies that the right hand side in the above formula tends to zero, which implies

lim k d ϕ,T (U A k , id) = lim k d ϕ,T (V A k , id) = 0.

Optimality of Belinskaya's theorem

We are now ready to prove Theorem 2.1.2: for any sublinear function ϕ, Belinskaya's theorem fails if we replace integrability by ϕ-integrability.

Theorem 2.4.14. -Let ϕ : R + → R + be a sublinear function and let T 1 ∈ Aut(X, µ) be ergodic. Then there exists an ergodic transformation T 2 ∈ [T 1 ] whose cocycle is ϕintegrable such that T 1 and T 2 have the same orbits but are not flip-conjugate.

The proof of the theorem depends on whether T 1 is weakly mixing. Indeed if it is the case, then we can use Corollary 2.3.11. Otherwise, we have to use the Baire category theorem. Indeed the candidate for T 2 is generic for the topology induced by d ϕ,T 1 .

Theorem 2.4.15. -Let ϕ be a sublinear metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic element. Then the set of all elements of [T] ϕ which are weakly mixing and have the same orbits as T is a dense G δ set in the Polish space of aperiodic elements of [T] ϕ with respect to the topology induced by d ϕ,T .

We delay the proof of the above theorem to Section 2.4.5. Let us first explain how to deduce Theorem 2.4.14 from Theorem 2.4.15.

Proof of Theorem 2.4.14. By Lemma 2.2.12, there is a sublinear metric-compatible function ψ such that ϕ(t) ≤ ψ(t) for all t large enough. In particular, ψintegrability implies ϕ-integrability for Z-valued functions (cf. Remark 2.2.13). Hence if the theorem holds for ψ then it holds for ϕ. Therefore, by replacing ϕ by ψ, we may and do assume that ϕ is a metric-compatible function.

If T 1 is weakly mixing, then all its nontrivial power are ergodic. Thus Corollary 2.3.11 implies that there exists T 2 ∈ [T 1 ] ϕ such that T 1 and T 2 have the same orbits but are not flip-conjugate.

If T 1 is not weakly mixing, then Theorem 2.4.15 grant us the existence of some weakly mixing T 2 ∈ [T 1 ] ϕ such that T 1 and T 2 have the same orbits. Since T 2 is weakly mixing and T 1 isn't, they cannot be flip-conjugated.

Weakly mixing elements form a dense G δ set

This section is dedicated to the proof of Theorem 2.4.15. Before starting the proof, we will need some terminology and preliminary propositions.

In this section we will consider the ϕ-integrable full groups both with the topology induced by the uniform metric d u and the their natural topology induced by d ϕ,T . The metric d ϕ,T is complete so we can apply the Baire category theorem in ([T] ϕ , d ϕ,T ), see Theorem 2.4.1. Moreover, the topology induced by d ϕ,T refines the topology induced by d u , see Corollary 2.2.18. Note that

([T] ϕ , d u ) is not complete, indeed one can show that [T] ϕ is dense in the com- plete metric space ([T], d u ).
Denote by APER ⊆ Aut(X, µ) the set of aperiodic transformations.

Lemma 2.4.16. -Let ϕ be a metric-compatible function and let T ∈ Aut(X, µ) be an aperiodic element. Then the set APER ∩ [T] ϕ is closed in the complete metric space ([T] ϕ , d ϕ,T ) and hence it is a complete metric space itself.

Proof. Note that T is aperiodic if and only if for all n ≥ 1 we have

d u (T n , id) = 1. So the set APER is closed in (Aut(X, µ), d u ). In particular, APER ∩ [T] ϕ is closed in ([T] ϕ , d u ), so it is also closed in ([T] ϕ , d ϕ,T ).
Proposition 2.4.17. -Let ϕ be a sublinear metric-compatible function and consider an aperiodic element T ∈ Aut(X, µ). Then the set

{U ∈ APER ∩ [T] ϕ : T and U have the same orbits} is a dense G δ set in (APER ∩ [T] ϕ , d ϕ,T ).
Proof. We first prove that this set is G δ . For all ε > 0 and n ≥ 1, let We now prove the density. Let U ∈ APER ∩ [T] ϕ . Fix a sequence (A k ) k≥0 of measurable subset of X which intersect every S-orbits, such that lim k µ(A k ) = 0 and lim k d ϕ,T (U A k , id) = 0 as in Proposition 2.4.13. If we set P k := (U A k ) -1 U, then we get that (P k ) k≥0 tends to U. Moreover, Corollary 2.3.7 implies that (T A k ) k≥0 tends to the identity, which implies that (T A k P k ) k≥0 tends to U. On the other hand, Lemma 2.2.4 yields that the transformation P k is periodic and A k is a fundamental domain for it. Thus, the transformations T A k P k and T have the same orbits by Lemma 2.2.3 and the proof is completed.

O ε,n := U ∈ [T] ϕ : µ({x ∈ X : T(x) ∈ {U -n (x), . . . , U n (x)}}) > 1 -ε . Each O ε,n is open in ([T] ϕ , d u
Let ERG denote the set of ergodic transformations in Aut(X, µ).

Proposition 2.4.18. -Let ϕ be a sublinear metric-compatible function and fix an ergodic transformation T ∈ Aut(X, µ).

Then ERG ∩ [T] ϕ is a dense G δ set in (APER ∩ [T] ϕ , d ϕ,T ). Proof. By [Kec10, Thm. 3.6], ERG ∩ [T] is a G δ set in (APER ∩ [T], d u ). Thus ERG ∩ [T] ϕ is a G δ set in (APER ∩ [T] ϕ , d ϕ,T ). Finally, since T is ergodic, ERG ∩ [T] ϕ ⊇ {S ∈ APER ∩ [T] ϕ : S and T have the same orbits}.
Thus, Proposition 2.4.17 yields that ERG ∩ [T] ϕ is dense in (APER ∩ [T] ϕ , d ϕ,T ) and the proof is complete.

Remark 2.4.19. -The hypothesis that ϕ is sublinear is necessary, as for any ergodic T ∈ Aut(X, µ), we have that ERG ∩ [T] 1 is not dense in APER ∩ [T] 1 . Indeed, one can define a continuous index map I : [T] 1 → Z by integrating the cocycle (see [LM18,Cor. 4.20] for the fact that it takes values in Z). Then note that I(U) = 0 for every ergodic U ∈ [T] 1 : by [LM18, Prop. 4.13] every ergodic U ∈ [T] 1 is either almost positive or almost negative. Then combining [LM18, Prop. 4.17 and Prop. 3.4] yields that U has positive or negative index, so I(U) = 0. Finally, there are aperiodic elements with index 0: take A ⊆ X measurable with 0 < µ(A) < 1, then the aperiodic transformation U := T A T -1 X\A has index zero (using again [LM18, Prop. 3.4]). By continuity of the discrete-valued index map, we conclude that ERG ∩ [T] 1 cannot be dense in APER ∩ [T] 1 .

Definition 2.4.20. -A transformation S ∈ Aut(X, µ) is weakly mixing if for all finite subset F ⊆ MAlg(X, µ) and all ε > 0, there exists n ∈ Z such that for all

A, B ∈ F , |µ(V n (A) ∩ B) -µ(A)µ(B)| < ε.
Given a measurable subset of positive measure A ⊆ X we will denote by µ A the probability measure on A defined by µ A (B) := µ(A ∩ B)/µ(A). We say that a transformation T ∈ Aut(X, µ) is weakly mixing on A if T(A) = A and the restriction of T to A is weakly mixing as an element of Aut(A, µ A ). The following result will be crucial in the proof of Theorem 2.4.15.

Theorem 2.4.21 (Conze [START_REF] Pierre | Équations fonctionnelles et systèmes induits en théorie ergodique[END_REF]). -Let T ∈ Aut(X, µ) be an ergodic transformation. Then the set

{A ∈ MAlg(X, µ) : T A is weakly mixing on A} is a dense G δ set in (MAlg(X, µ), d µ ) where d µ (A, B) := µ(A B).
Denote by WMIX the set of weakly mixing transformations of Aut(X, µ). We are finally ready to prove Theorem 2.4.15 which can be reformulated as follows.

Theorem 2.4.22. -Let ϕ be a sublinear metric-compatible function and let T ∈ Aut(X, µ) be an ergodic transformation. Then the set

{U ∈ WMIX ∩ [T] ϕ : T and U have the same orbits} is a dense G δ set in (APER ∩ [T] ϕ , d ϕ,T ).
Proof. By the Baire category theorem, the intersection of two dense G δ subsets is a dense G δ subset. Hence by Proposition 2.4.17, it suffices to show that WMIX ∩ [T] ϕ is a dense G δ set in (APER ∩ [T] ϕ , d ϕ,T ), which will occupy the remainder of the proof.

By definition, a transformation U is weakly mixing if and only if for all finite subset F ⊆ MAlg(X, µ) and all ε > 0, it belongs to the set O F ,ε defined by:

O F ,ε := {V ∈ Aut(X, µ) : ∃n ∈ Z, ∀A, B ∈ F , |µ(V n (A) ∩ B) -µ(A)µ(B)| < ε} . Observe that each O F ,ε is open in (Aut(X, µ), d u ). As before, denote by d µ the metric on MAlg(X, µ) defined by d µ (A, B) = µ(A B).
Claim 1. -Let F = {A 1 , ..., A n } and F = {A 1 , ..., A n } be subsets of MAlg(X, µ). Fix ε > 0. If for every i ∈ {1, . . . , n} one has µ(

A i A i ) < ε, then O F ,ε ⊆ O F ,5ε . Proof of the claim. Let V ∈ O F ,ε . Fix n ∈ Z such that for all i, j ∈ {1, . . . , n} we have |µ(V n (A i ) ∩ A j ) -µ(A i )µ(A j )| < ε.
We first remark that for every measurable B ⊂ X and i ∈ {1, . . . , n}, we have |µ(B)µ

(A i ) -µ(B)µ(A i )| < ε and |µ(B ∩ A i ) -µ(B ∩ A i )| < ε.
The result now follows from the triangular inequality and the fact V preserves µ:

|µ(V n (A i ) ∩ A j ) -µ(A i )µ(A j )| < |µ(V n (A i ) ∩ A j ) -µ(A i )µ(A j )| + ε < |µ(V n (A i ) ∩ A j ) -µ(A i )µ(A j )| + 2ε < |µ(V n (A i ) ∩ A j ) -µ(A i )µ(A j )| + 3ε = |µ(A i ∩ V -n (A j )) -µ(A i )µ(A j )| + 3ε < |µ(A i ∩ V -n (A j )) -µ(A i )µ(A j )| + 4ε = |µ(V n (A i ) ∩ A j ) -µ(A i )µ(A j )| + 4ε < 5ε. claim Since (X, µ) is standard, we can fix a countable dense subset M of (MAlg(X, µ), d µ ). It follows from the Claim 1 that WMIX = ε∈Q * + F ⊆M finite O F ,ε . (2.7) In particular WMIX is a G δ set in (Aut(X, µ), d u ) and hence WMIX ∩ [T] ϕ is a G δ set in ([T] ϕ , d ϕ,T ).
We now prove that WMIX is dense. By the Baire category theorem, it is enough to show that each O

F ,ε is dense in (APER ∩ [T] ϕ , d ϕ,T ). By Proposition 2.4.18 the set ERG ∩ [T] ϕ is dense in APER ∩ [T] ϕ , so it is enough to prove that ERG ∩ [T] ϕ ⊆ O F ,ε ∩ APER ∩ [T] ϕ .
(2.8) So let us fix a finite subset F ⊆ MAlg(X, µ), a positive real ε > 0 and an ergodic transformation U ∈ ERG ∩ [T] ϕ .

We let (X k ) k≥0 be a sequence of measurable subsets such that µ(X k ) = 1 -2 -k . For all k ≥ 0, we apply Conze's Theorem to the transformation U X k , which is ergodic on X k , to find a measurable subset

Y k ⊆ X k such that µ(Y k ) > 1 - 2 -k+1 and U Y k is weakly mixing on Y k . Set V k := U Y k T X\Y k . We claim that (V k ) k≥0 tends to U. Indeed since lim k µ(Y k ) = 1, Proposition 2.

2.A Proof of Belinskaya's theorem

In this appendix, we present a short proof of Belinskaya's theorem due to Katznelson which is not publicly available to our knowledge. As in Belinskaya's original proof, a key step is the following theorem, of independent interest. The proof we present here is mainly due to Katznelson. To lighten notation, given a point x ∈ X, a map T : X → X, and a subset I ⊆ Z, we will write

T I (x) := {T i (x) : i ∈ I}.
Theorem 2.A.1. -Let T be an aperiodic measure-preserving transformation, suppose U ∈ Aut(X, µ) has the same orbits as T and that for almost every x ∈ X, the symmetric difference of the respective positive T and U orbits T N (x) U N (x) is finite. Then T and U are conjugate. Proof (Katznelson). We will explicitly define an element S in [T] such that U = S -1 TS. This will be done thanks to the following claim.

Claim. -For almost every x ∈ X, there exists a unique j(x) ∈ Z such that

T N+j(x) (x) \ U N (x) = U N (x) \ T N+j(x) (x)
Proof of the claim. For almost every x ∈ X, consider the function τ x : Z → Z defined by

τ x (j) := T N+j (x) \ U N (x) -U N (x) \ T N+j (x) .
Remark that by assumption for almost every x, the value τ x (j) is finite for all j ∈ Z. By considering the two cases T j (x) ∈ U N (x) and T j (x) ∈ U N (x), we see that τ x (j + 1) = τ x (j) + 1 for all j ∈ Z. It follows that τ x (j) = τ x (0) + j for all j ∈ Z so j(x) := -τ x (0) is the unique element we seek. claim We set S(x) := T j(x) (x). By the above claim, S(x) is the unique element of the T-orbit of x satisfying

T N (S(x)) \ U N (x) = U N (x) \ T N (S(x)) .
(2.9) By considering whether none, only one, or both of the points x and S(x) belong to T N (S(x)) ∩ U N (x), we see that removing the point S(x) from T N (S(x)) and the point x from U N (x) does not perturb the above equation, so that

T N+1 (S(x)) \ U N+1 (x) = U N+1 (x) \ T N+1 (S(x)) .
This can be rewritten as

T N (TS(x)) \ U N (U(x)) = U N (U(x)) \ T N (TS(x)) ,
which by equation (2.9) yields the desired equivariance condition SU(x) = TS(x).

We now have to check that S ∈ [T]. Using that T and U are invertible and a straightforward induction, we obtain that SU n (x) = T n S(x) for all n ∈ Z. In particular S induces a bijection from the T-orbit of x to the U-orbit of S(x). Since S(x) = T j(x) (x) belongs to the T-orbit of x, which coincides with the U-orbit of x, we conclude that S induces a bijection on each T-orbit, in particular S is bijective. Finally we check that S is measure-preserving. The sets A n := {x ∈ X : S(x) = T n (x)} for n ∈ Z form a partition of X. If B ⊆ X is measurable, we write

B = n A n ∩ B so that µ(S(B)) = ∑ n µ(T n (A n ∩ B)) = ∑ n µ(A n ∩ B) = µ(B).
This ends the proof of Theorem 2.A.1

Given T ∈ Aut(X, µ), denote by R T ⊆ X × X the equivalence relation whose classes are the T-orbits. Before proceeding with the proof of Belinskaya's theorem, we recall the following well-known lemma. Its usefulness towards proving Belinskaya's theorem was pointed out to us by Todor Tsankov. Lemma 2.A.2 (Mass-transport principle). -Let T ∈ Aut(X, µ) and f : R T → N be a measurable map. Then

X ∑ n∈Z f (x, T n (x))dµ = X ∑ n∈Z f (T n (x), x)dµ.
Proof. Since f is non-negative, Tonelli's theorem tells us that Proof. Define a T-invariant total order ≤ T on each T-orbit by setting x ≤ T y if there is n ≥ 0 such that y = T n (x). We will write x < T y whenever x = y and x ≤ T y. Define f : R T → N by:

X ∑ n∈Z f (x, T n (x))dµ = ∑ n∈Z X f (x, T n (x))dµ = ∑ n∈Z X f (T -n (x), x)dµ = ∑ n∈Z X f (T n (x), x)dµ = X ∑ n∈Z f (T n (x), x)dµ.
f (x, y) := 1 if x ≤ T y < T U(x) or U(x) < T y ≤ T x, 0 otherwise. Let us denote by c U the T-cocycle of U. By assumption, c U is integrable. Re- mark that f (x, T n (x)) = 1 if and only if 0 ≤ n < c U (x) or c U (x) < n ≤ 0, so ∑ n∈Z f (x, T n (x)) = |c U (x)|. We thus have X ∑ n∈Z f (x, T n (x))dµ = X |c U (x)| dµ < +∞.
Using the mass-transport principle (Lemma 2.A.2), we deduce that

X ∑ n∈Z f (T n (x), x)dµ < +∞,
in particular for almost every x ∈ X, the sum ∑ n∈Z f (T n (x), x) is finite.

This implies that for almost every x ∈ X, there are only finitely many integers n ∈ Z such that U n (x) ≤ T x < T U n+1 (x) or U n+1 (x) < T x ≤ T U n (x). Since the U-orbit of almost every point is infinite, for almost every x, we must have that either U n (x) ≤ T x or U n (x) ≥ T x for all but finitely many n ≥ 0. By ergodicity of U and up to replacing U with its inverse, we can assume that for almost all x ∈ X we have U n (x) ≥ T x for all but finitely many n ≥ 0.

By the symmetric argument, for all but finitely many n ≤ 0, we have that U n (x) ≤ T x and therefore we must have that {T n (x) : n ≥ 0} {U n (x) : n ≥ 0} is finite. The conclusion now follows from Theorem 2.A.1.

Chapter 3

Cycles in ϕ-integrable full groups This chapter is devoted to exploring consequences of the shift-coupling inequality in the context of ϕ-integrable full groups. This inequality is due to Thorisson [START_REF] Thorisson | Coupling methods in probability theory[END_REF] and was used by Liggett to prove the following probabilistic theorem.

Theorem 3.0.1 ([Lig02, Thm. 1.1]). -Let (X n ) n∈Z be a sequence of i.i.d. random variables whose distribution is the Bernoulli distribution of parameter 1/2. Let µ be the conditional law of (X n ) n∈Z given that X 0 = 1. Then any Z-valued random variable N such that the law of (X n+N ) n∈Z is equal to

µ satisfies E[|N| 1/2 ] = ∞.
The proof provided by Liggett in [Lig02] is probabilistic. We propose here to follow the lines of Liggett's proof with the language of ϕ-integrable full groups introduced in [CJLMT22]. The aim is to state Theorem 3.0.1 in terms of cycles in ϕ-integrable full groups of Bernoulli shifts, see Theorem 3.2.3. Apart from the shift of point of view and the use of new objects such as ϕ-integrable full groups, no originality is claimed in this chapter.

Shift-coupling inequality

The inequality

Let X be a standard Borel space. If µ and ν are two Borel measures on X, we denote by µν TV the total variation distance between µ and ν, defined by

µ -ν TV := sup X f (x)dµ - X f (x)dν , f : X → [-1, 1] measurable .
Let µ and ν be two Borel measures. Then ν is absolutely continuous with respect to µ if for all Borel subset A ⊆ X, µ(A) = 0 implies ν(A) = 0. If ν is absolutely continuous with respect to µ and µ, ν are σ-finite measures, then Radon-Nikodym Theorem states that there exists a function dν/dµ : X → R + called the Radon-Nikodym derivative of ν with respect to µ, which is unique up to equality µ-almost everywhere, such that for all A ⊆ X,

ν(A) = A dν dµ (x)dµ.
There is a useful formula to compute the total variation distance between two measures when one is absolutely continuous with respect to the other.

Lemma 3.1.1. -Let µ and ν be two σ-finite Borel measures on X. If ν is absolutely continuous with respect to µ, then

µ -ν TV = X 1 - dν dµ (x) dµ.
Proof. For any measurable function f :

X → [-1, 1], X f (x)dµ - X f (x)dν = X f (x) 1 - dν dµ (x) dµ ≤ X 1 - dν dµ (x) dµ.
Since the last quantity is independent of f , we then get that

µ -ν TV ≤ X 1 - dν dµ (x) dµ.
For the reverse inequality, let f : X → [-1, 1] be defined by f

(x) = 1 if (dν/dµ)(x) ≤ 1 and f (x) = -1 if (dν/dµ)(x) > 1. Then we obtain µ -ν TV ≥ X f (x)dµ - X f (x)dν = X f (x) 1 - dν dµ (x) dµ = X 1 - dν dµ (x) dµ,
which proves the lemma.

In the sequel, if T : X → X is a Borel bijection and c : X → Z a Borel map, we denote by T c : X → X the map defined by T c (x) := T c(x) (x).

Proposition 3.1.2 (Shift-coupling inequality). -Let T : X → X be a Borel bijection. Let µ be a Borel measure on X and let c : X → Z be a Borel map. For all n ≥ 1, let ν n be the measure defined by

ν n := T * µ + • • • + (T n ) * µ. Then ν n -(T c ) * ν n TV ≤ 2 X min(|c(x)|, n)dµ.
Proof. If k ≤ l are two integers, we denote by [[k, l]] the interval {k, k + 1, . . . , l} and by 1 A the indicator function of a set A. Let f : X → [-1, 1] be a measurable function. Then we have

X ( f (x) -f (T c(x) (x)))dν n = n ∑ k=1 X f (T k (x))dµ - n ∑ k=1 X f (T k+c(x) (x))dµ = ∑ k≥0 X f (T k (x))(1 [[1,n]] (k) -1 [[c(x)+1,c(x)+n]] (k))dµ ≤ ∑ k≥0 X |1 [[1,n]] (k) -1 [[c(x)+1,c(x)+n]] (k)|dµ.

Shift-coupling inequality and ϕ-integrability

With the same notations as in Proposition 3.1.2, our goal in the sequel is to prove that if the function c : X → Z satisfies some kind of integrability condition, then it forces the behavior of ν n -(T c ) * ν n TV to be slow as n goes to +∞. The integrability conditions in question are obtained by metric-compatible function.

Definition 3.1.3. -A function ϕ : R + → R + is metric-compatible if:

• (subadditivity) for all s, t ∈ R + , ϕ(s + t) ≤ ϕ(s) + ϕ(t).

• (separation) ϕ(0) = 0 and ϕ(t) > 0 for all t > 0.

• (monotonicity) ϕ is a non-decreasing function.

A metric-compatible function

ϕ : R + → R + is sublinear if lim t→+∞ ϕ(t)/t = 0.
Observe that if ϕ is a metric-compatible function and d is a metric on some set, then ϕ • d is still a metric. Examples of metric-compatible functions are concave functions ϕ : R + → R + such that ϕ(0) = 0.

The following lemma is a useful technicality on metric-compatible functions.

→ R is ϕ-integrable if X ϕ(| f (x)|)dµ < +∞.
Proposition 3.1.6. -Let ϕ : R + → R + be a sublinear metric-compatible function.

Let T : X → X be a Borel bijection and µ a Borel measure on X. For all n ≥ 1, let ν n be the measure defined by

ν n := T * µ + • • • + (T n ) * µ. If c : X → Z is any ϕ-integrable map, then lim n ϕ(n) n ν n -(T c ) * ν n TV = 0.
Proof. For all x ∈ X and n ≥ 1, let c n (x

) := ϕ(n) n min(|c(x)|, n). The sequence of positive functions (c n ) n≥0 converges pointwise to 0. Let n ≥ 1. If x ∈ X is such that |c(x)| > n, then c n (x) = ϕ(n) ≤ ϕ(|c(x)|). If x ∈ X is such that 0 < |c(x)| ≤ n, then we have by Lemma 3.1.4 c n (x) = |c(x)| ϕ(n) n = ϕ(|c(x)|) ϕ(n) n |c(x)| ϕ(|c(x)|) ≤ 2ϕ(|c(x)|).
Thus, for all x ∈ X, we obtain that c n (x) ≤ 2ϕ(|c(x)|). Lebesgue's dominated convergence theorem applied to the sequence (c n ) n≥0 yields

lim n ϕ(n) n X min(|c(x)|, n)dµ = 0.
Finally, the shift-coupling inequality of Proposition 3.1.2 leads to the desired conclusion.

Application to ϕ-integrable full groups

In this section, we apply the shift-coupling inequality, or more precisely its consequence obtained in Proposition 3.1.6 to get constraints on the existence of cycles with prescribed properties in ϕ-integrable full groups. Let us first give the definition of these groups, which were introduced in [CJLMT22]. Let T ∈ Aut(X, µ) be an aperiodic transformation. We denote by [T] the full group of T, that is, the group of all measure-preserving transformations U ∈ Aut(X, µ) such that for all x ∈ X, there exists n ∈ Z such that U(x) = T n (x). Any U ∈ [T] is completely determined by its T-cocycle, defined as the unique function c U : X → Z satisfying the equation U(x) = T c U (x) (x) for all x ∈ X.

Definition 3.2.1. -Let ϕ : R + → R + be a metric-compatible function. The Thus, for all n ≥ 1, we have

C ≤ E n -mS n √ n .
Moreover, by definition of S n , we have

E n -mS n √ n = 1 √ n X n -m n ∑ k=1 1 D (T -k (x)) dµ.
Finally, Theorem 3.2.2 implies that lim n ϕ(n) √ n = 0, as claimed.

Remark 3.2.4. -If ϕ is a metric-compatible function such that ϕ(n)/ √ n → 0, Theorem 3.2.3 yields that there is no m-cycle P in the ϕ-integrable full group of the Bernoulli shift with base ({1, . . . , m}, κ) and with fundamental domain

{(x n ) n∈Z : x 0 = a}.
When m = 2, this result is optimal. In this case, one can construct an explicit involution which belongs to the L p full group of the Bernoulli shift for all p < 1/2, as follows. Let T be the Bernoulli shift on ({0, 1}, κ) ⊗Z , where κ is the uniform measure on {0, 1}. Let X 0 := {(x n ) n∈Z ∈ {0, 1} Z : x 0 = 0}. For all x ∈ X 0 , let N(x) be the infimum of n ≥ 1 such that 1 appears strictly more often than 0 in {x 1 , . . . , x n }. Then the map π : x ∈ X 0 → T N(x) (x) ∈ {0, 1} Z \ X 0 is well-defined and injective almost everywhere. Thus it can be extended to an involution P ∈ [T] with full support and fundamental domain X 0 . Standard estimates on the simple random walk on Z imply that P belongs to [T] p for all 0 < p < 1/2. 4.4, we use this combinatorial result and the existence of non-crossing involution in L <1 full-groups of ergodic transformations to prove item (i) of Theorem 4.0.1.

Preliminaries

Let Γ and Λ be two countable groups. Let Γ (X, µ) be a p.m.p. action. A cocycle into Λ is a map σ : Γ × X → Λ such that for all γ, δ ∈ Γ, x ∈ X,

σ(γδ, x) = σ(γ, δ • x)σ(δ, x).
Cocycles naturally appear with orbit equivalent actions. Two p.m.p. essentially free actions Γ (X, µ) and Λ (Y, ν) are orbit equivalent if there exists an orbit equivalence between them, that is, a bimeasurable bijection Φ : X → Y such that Φ * µ = ν and for µ-almost every x ∈ X, we have

Φ(Γ • x) = Λ • Φ(x).
The map Φ is called an orbit equivalence between the actions. By freeness of the actions, there are two measurable maps σ : Γ × X → Λ and τ : Λ × Y → Γ, which are called the orbit equivalence cocycles associated with Φ and are defined for all γ ∈ Γ, λ ∈ Λ and µ-almost every x ∈ X by

Φ(γ • x) = σ(γ, x) • Φ(x), Φ(τ(λ, Φ(x)) • x) = λ • Φ(x).
Assume that Γ and Λ are finitely generated groups and fix |•| Γ and |•| Λ word lengths associated with finite generating systems for Γ and Λ respectively. We say that an orbit equivalence Φ : (X, µ) → (Y, ν) is an L p orbit equivalence (p ∈]0, +∞[) if the following two conditions are satisfied:

(i) X |σ(γ, x)| p Γ dµ < +∞ for all γ ∈ Γ, (ii) X |τ(λ, x)| p Λ dµ < +∞ for all λ ∈ Λ.
We say that Φ is an L ∞ orbit equivalence if for all γ ∈ Γ and λ ∈ Λ, the maps |σ(γ, -)| Γ et |τ(λ, -)| Λ are essentially bounded. In both cases, the conditions are independent of the choice of the word length |•| Γ and |•| Λ associated with finite generating systems. Moreover, the cocycle identity implies that it is enough to check these integrability conditions for γ and λ belonging to any finite generating system of Γ and Λ respectively.

We say that two p.m.p. actions Γ (X, µ) and Λ (Y, ν) are L p orbit equivalent (p ∈]0, +∞]) if there exists an L p orbit equivalence between them. Finally, we say that the p.m.p. actions are L <p orbit equivalent if there exists an orbit equivalence which is an L q orbit equivalence for all q < p.

L 1 orbit equivalence between Z and D ∞

Let D ∞ be the infinite dihedral group which is defined as the free product of two copies of the cyclic group of order two. A presentation of D ∞ is given by

D ∞ := a, b | a 2 = b 2 = 1 .
Our aim in this section is to understand when an ergodic action of Z can be L 1 orbit equivalent to an ergodic action of D ∞ . Proposition 4.2.1. -Let Z α (Y, ν) be an ergodic action. If there exists a p.m.p. essentially free action D ∞ β (X, µ) such that α and β are L 1 orbit equivalent, then the subgroup 2Z doesn't act ergodically on (X, µ).

Proof. Let Φ : (X, µ) → (Y, ν) be an L 1 orbit equivalence between α and β. Let σ : Z × X → D ∞ and τ : D ∞ × Y → Z be the orbit equivalence cocycles associated with Φ. Let us define τ : D ∞ × Y → Z by τ(γ, y) = -τ(γ -1 , y). By Bowen's theorem on extension of L 1 orbit equivalence cocycles on the space of ends [Bow17, Thm. 3.1], τ extends to a continuous map

τ : D ∞ ∪ {ξ ab , ξ ba } × Y → Z ∪ {-∞, +∞}
where ξ ab = (ab) ∞ , ξ ba = (ba) ∞ are the two ends of D ∞ and -∞, +∞ the two ends of Z. The map τ satisfies the extended cocycle identity: for all γ ∈ D ∞ , ξ ∈ D ∞ ∪ {ξ ab , ξ ba } and for ν-almost every y ∈ Y τ(γξ, y) = τ(γ, y) + τ(ξ, β(γ -1 )y).

Moreover, for ν-almost every y ∈ Y, we have { τ(ξ ab , y), τ(ξ ba , y)} = {-∞, +∞}. In our case, the action of Z on its space of ends {-∞, +∞} is trivial, thus for ν-almost every y ∈ Y and all γ ∈ D ∞ , we have τ(γξ ab , y) = τ(ξ ab , β(γ -1 )y) and τ(γξ ba , y) = τ(ξ ba , β(γ -1 )y).

Therefore, for ν-almost every y ∈ Y, τ(ξ ab , β((ab) -1 )y) = τ(ξ ab , y). there exists a maximal interval of the form [[t, P(t)]] which contains z and is contained in [[x, y]]. Since P is non-crossing, intervals of this form are either disjoint, or equal by maximality. Thus, there are x 1 , . . . ,

This proves that the measurable sets

x k ∈ [[x, y]] such that [[x, y]] = [[x 1 , P(x 1 )]] • • • [[x k , P(x k )]].
By permuting the elements x 1 , . . . , x k and changing x i to P(x i ) if necessary, we may assume that

x = x 1 < P(x 1 ) < • • • < x k < P(x k ) = y.

A combinatorial result

Let P : Z → Z be a fixed-point free, non-crossing involution. By Lemma 4.3.3, the map TPT -1 is a fixed-point free, non-crossing involution. Thus we can represent it with non-crossing circular arcs but this time drawn in the lower half-plane. We refer to these arcs as the TPT -1 lower arcs. Observe that the TPT -1 lower arcs are obtained as the image of the P upper arcs by a reflexion across the horizontal line, followed by the translation T : x → x + 1. Proof. Let G be the graph with vertex set Z and egde set {P upper arcs} ∪ {TPT -1 lower arcs}. We define the length of an arc to be the distance between its extremities.

Claim. -For x ∈ Z such that a • xx ≥ 1, there exists an integer n ≥ 1 such that (ba

) n • x = a • x + 1 and {x, a • x, (ba) • x, (aba) • x, . . . , (ba) n • x} = [[x, a • x + 1]].
This claim says that for such a point x, the path in the graph G which starts at x, then takes a P upper arc, then a TPT -1 lower arc, etc., will visit exactly every point in the interval [[x, a • x]] and then reach a • x + 1.

Proof of the claim. We prove this claim by induction on the length l of the P upper arc with extremities x and a • x, which is odd by Lemma 4.3.4. If l = 1, then n = 1 works. Assume that the result of the claim holds for all P upper arcs of length < 2d + 1. Let us prove that the result holds for any P upper arc of length l := 2d + 1. Let x ∈ Z such that a • xx = l. We apply Lemma 4.3.5 to the interval

[[x + 1, a • x -1]], which is P-invariant. There is x 1 , . . . , x k ∈ Z such that x + 1 = x 1 < P(x 1 ) < • • • < x k < P(x k ) and [[x + 1, a • x -1]] = [[x 1 , P(x 1 )]] • • • [[x k , P(x k )]].
The P upper arcs with extremities x i and P(x i ) are the P upper arcs located immediately below the P upper arc with extremities x and a • x. The length of any of these arcs is < l, thus one can use the induction hypothesis. There exists integers n 1 , . . . , n k ≥ 1 such that for all 1 ≤ i ≤ k, we have (ba

) n i • x i = a • x i + 1 and {x i , a • x i , (ba) • x i , . . . , (ba) n i • x} = [[x i , a • x i + 1]].
By concatenating these results, we obtain the claim with n

:= n 1 + • • • + n k + 1. claim
In order to prove that the graph G is connected, it is enough to prove that for all x ∈ Z, the points x and x + 1 are connected. Let x ∈ Z. In both of the cases a • xx ≥ 1 and xa • x ≥ 1, the fact that x and x + 1 are connected is a direct consequence of the claim. Thus, G is connected, which proves the theorem.

L <1 orbit equivalence between Z and D ∞

Let (X, µ) be a probability space. Let Aut(X, µ) be the group of all measurepreserving transformations of (X, µ). The support of an element T ∈ Aut(X, µ) is the set supp(T) := {x ∈ X : T(x) = x}.

Let T ∈ Aut(X, µ) be ergodic. The full group of T, denoted by [T], is the group of all bimeasurable bijections U : X → X such that U * µ = µ and for all x ∈ X, there exists n x ∈ Z such that U(x) = T n x x. By ergodicity of T, every T-orbit is infinite and thus the integer n x is unique. The map x → n x is called the T-cocycle of U and is denoted by c U . The L 1 full group of T, denoted by [T] 1 is the group of all

U ∈ [T] such that X |c U (x)|dµ < +∞.
This group was introduced by Le Maître who proved that this is a complete invariant of flip-conjugacy for ergodic actions of Z [LM18].

Chapter 4

Thus, for all 0 < p < 1, we have

X |c P (x)| p dµ = µ(A 1 \ A 2 ) + ∑ k≥2 A k \A k+1 |c P (x)| p-1 |c P (x)|dµ ≤ µ(A 1 \ A 2 ) + ∑ k≥2 2 (k-1)(p-1) A k \A k+1 |c P (x)|dµ.
Claim. -For all k ≥ 1, we have

A k \A k+1 |c P (x)|dµ ≤ 2.
Proof of the claim. For all k ≥ 1, we define

L k := {x ∈ A k : n k (x) is even and n k (T k (x)) = 0}. Set R k := P(L k ). Then A k \ A k+1 = L k R k .
Moreover, for all x ∈ L k we have c P (x) = n T,A k (x) and for all x ∈ R k we have c P (x) = -n T,A k (P(x)). Thus

A k \A k+1 |c P (x)|dµ = 2 L k n T,A k (x)dµ ≤ 2 A k n T,A k (x)dµ = 2,
where the last equality is given by Kac's lemma.

claim Therefore, we get that X |c P (x)| p dµ is finite for every 0 < p < 1, which proves that P ∈ [T] <1 .

If along the above construction we get µ(A n ) = 0 for some n ≥ 1, then we define P := P 1 • • • P n . By the same computation, this yields a non-crossing involution with full support in [T] <1 . Theorem 4.4.4. -Let Z α (X, µ) be an ergodic action. Then there exists an ergodic action D ∞ β (Y, ν) such that α and β are L <1 orbit equivalent.

Proof. Let T := α(1). By Theorem 4.4.3, let P ∈ [T] <1 be a non-crossing involution with full support. Then TPT -1 is also a non-crossing involution with full support, which belongs to [T] <1 . Let D ∞ β (X, µ) be the p.m.p. action defined by β(a)x := P(x) and β(b)x := TPT -1 (x). By Theorem 4.3.6, for µ-almost every x ∈ X, the restriction of β to the T-orbit of x is transitive. This proves that id X is an orbit equivalence between α and β. Moreover, the fact that P and TPT -1 belong to [T] <1 implies right away that α and β are L <1 orbit equivalent.

It is unclear how to prove a "dual" result to Theorem 4.4.4. That is, given an ergodic action D ∞ α (X, µ), is there a way to construct an ergodic action Z β (Y, ν) such that α and β are L <1 orbit equivalent?

Part III Quantitative orbit equivalence and graphings of bounded degree graphs, see [Lov12,Part 4] for an introduction to this theory. They are also one of the main objects in the cost theory of p.m.p. actions, which was extensively studied by Gaboriau [Gab00]. Graphings are closely related to probability measure preserving actions of countable groups. A probability measure preserving action (p.m.p. action for short) Γ α (X, µ) of a countable group Γ on a standard probability space (X, µ) is a collection (α(γ)) γ∈Γ of bimeasurable bijections α(γ) : X → X which preserve the probability measure µ, such that for all γ, δ ∈ Γ, we have α(γδ) = α(γ)α(δ). A p.m.p. action is essentially free if the set of points with trivial stabilizer has full measure. A p.m.p. action is ergodic if any measurable set, which is invariant under the action, has measure 0 or 1. Assume that Γ is finitely generated and fix S a finite generating system for Γ, that is, a finite symmetric set which generates the group and which does not contain the identity element 1 Γ . To any p.m.p. action Γ α (X, µ), one can associate a graphing, denoted by α(S Γ ), whose vertex set is X, and whose edge set is the symmetric set {(x, x ) ∈ X × X : ∃s ∈ S, α(s)x = x }.

The spirit of this article fits into the framework of quantitative orbit equivalence and more generally quantitative measure equivalence, [DKLMT20]. These nascent areas aim to understand how metric structures on p.m.p. actions are distorted under orbit and measure equivalences.

Bounded orbit equivalence appears to be an important notion of quantitative orbit equivalence. For instance, in the entropy theory, Austin proved that any two p.m.p. essentially free actions of amenable groups that are bounded orbit equivalent (and even integrable orbit equivalent) have the same Kolmogorov-Sinai entropy [START_REF] Austin | Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence[END_REF]. Bowen and Lin proved that any two p.m.p. essentially free actions of a free group of finite rank, which are bounded orbit equivalent, have the same f -invariant [START_REF] Bowen | Entropy for actions of free groups under bounded orbit-equivalence[END_REF]. Let Γ and Λ be two finitely generated groups and fix two finite generated systems S Γ and S Λ for Γ and Λ respectively. We say that two p.m.p. actions Γ α (X, µ) and Λ β (Y, ν) are bounded orbit equivalent if there exists a constant C > 0 and a bimeasurable bijection Φ : X → Y such that Φ * µ = ν and for µ-almost every x ∈ X, the map Φ is a C-biLipschitz map between the connected component of x ∈ X in the graphing α(S Γ ) and the connected component of Φ(x) ∈ Y in the graphing β(S Λ ). It is straightforward to check that bounded orbit equivalence does not depend on the choice of finite generating systems for the groups.

This notion retains the geometry of the group. For instance, the following result can be extracted from Shalom's work [START_REF] Shalom | Harmonic analysis, cohomology, and the largescale geometry of amenable groups[END_REF]: two finitely generated amenable groups Γ and Λ admit p.m.p. essentially free actions that are bounded orbit equivalent if and only if Γ and Λ are biLipschitz equivalent. This result is very specific to the amenable realm, as for instance free groups of different finite ranks are biLipschitz equivalent [START_REF] Papasoglu | Homogeneous trees are bilipschitz equivalent[END_REF] but not bounded orbit equivalent because they are not even orbit equivalent [Gab00]. While bounded orbit equivalence is related to the study of graphings which are uniformly biLipschitz, we propose here to study graphings associated with p.m.p. actions up to isometry. Definition 5.1.1. -Let Γ and Λ be two finitely generated groups and fix two finite generating systems S Γ and S Λ respectively. Two p.m.p. actions Γ α (X, µ) and Λ β (Y, ν) are isometric orbit equivalent if the graphings α(S Γ ) and β(S Λ ) are isometric in a measurable sense, that is, if there exists a bimeasurable bijection Φ : X → Y such that Φ * µ = ν and for µ-almost every x ∈ X, the map Φ is an isometry between the connected component of x ∈ X in the graphing α(S Γ ) and the connected component of Φ(x) ∈ Y in the graphing β(S Λ ).

Beware that contrary to bounded orbit equivalence, the notion of isometric orbit equivalence heavily depends on the choice of generating systems. In each of the statements in this article, when we say that two actions are isometric orbit equivalent, it refers to the generating systems of the groups that are fixed and clear in the context.

Consider Γ a finitely generated group and fix S Γ a finite generating system of it. The Cayley graph (Γ, S Γ ) is the graph whose vertex set is Γ and whose edge set is the symmetric set {(γ, δ) ∈ Γ 2 : ∃s ∈ S Γ , δ = γs}. We endow the Cayley graph with its graph distance. Two Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric if there exists a bijective isometry between them. Our first result shows that the existence of isometric orbit equivalent actions that are essentially free is connected to the isometry class of the Cayley graph.

Theorem 5.1.2 (see Theorem 5.3.3). -Let Γ and Λ be two finitely generated groups and fix two finite generating systems S Γ and S Λ respectively. Then Γ and Λ admit p.m.p. essentially free actions that are isometric orbit equivalent if and only if the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric.

One direction of this result is immediate, because in the graphing associated with any p.m.p. essentially free action, almost every connected component is isometric to the Cayley graph of the acting group. Thus, an isometric orbit equivalence between p.m.p. essentially free actions leads to an isometry between the Cayley graphs on almost every connected components of the associated graphings. The converse is proved using the space Iso(Γ, Λ) of bijective isometries from (Γ, S Γ ) to (Λ, S Λ ). The group Λ acts by postcomposition on Iso(Γ, Λ). Similarly, the group Γ acts by postcomposition on the space Iso(Λ, Γ) of bijective isometries from Λ to Γ. We prove that the quotient actions Γ Iso(Γ, Λ)/Λ and Λ Iso(Λ, Γ)/Γ, endowed with their respective Haar probability measures, are isometric orbit equivalent, see Corollary 5.3.2. This connection between Cayley graph and isometric orbit equivalence leads to interesting rigidity results. Given a finitely generated group Γ with a generating system S Γ , we denote by Iso(Γ) the group of bijective isometries of the Cayley graph (Γ, S Γ ). Equivalently, Iso(Γ) is isomorphic to the group of all graph automorphisms of the Cayley graph. We prove the following result using techniques introduced by Furman in [Fur99].

Theorem 5.1.3 (see Theorem 5.4.1). -Let Γ and Λ be two finitely generated groups and fix two finite generating systems S Γ and S Λ respectively. Assume that the Cayley equivalent to the diagonal action Γ Iso(Γ)/Γ × Γ/Λ, equipped with the product of the Haar probability measure and the uniform probability measure. The aim in order to prove Theorem 5.1.4 becomes to show that for some subgroups Λ ≤ F d , the action F d Iso(F d )/Λ is mixing. We actually give a complete characterization of such subgroups Λ in Section 5.5.2. Along the proof, we use the Howe-Moore property satisfied by the group Iso(F d ), a result due independently to Lubotzky and Mozes [START_REF] Lubotzky | Asymptotic Properties of Unitary Representations of Tree Automorphisms[END_REF] and to Pemantle [START_REF] Pemantle | Automorphism invariant measures on trees[END_REF].

Preliminaries

Cayley graph

In the text, if we write Γ = S , we always mean that Γ is a countable group and S a generating system for Γ, that is, a symmetric set which generates Γ, and which does not contain the identity element 1 Γ of Γ. In the case where the generating system S is finite, then we say that Γ is finitely generated. When we say that Γ = S is finitely generated, we implicitly mean that S is a finite generating system.

Let Γ = S be a countable group. We denote by |•| S the word length on Γ with respect to the generating system S. It is defined by |1 Γ | S = 1 and for all γ ∈ Γ \ {1 Γ }, |γ| S := min{n ∈ N : ∃s 1 , . . . , s n ∈ S, γ = s 1 . . . s n }.

The Cayley graph of Γ = S , denoted (Γ, S), is the simplicial graph whose vertex set is Γ endowed with the metric (γ, δ) → |γ -1 δ| S , called the word metric. In practice, this means that (Γ, S) is a graph without multiple edges or loops and there is an edge between two vertices γ, δ ∈ Γ if and only if there is s ∈ S such that γ = δs. A map f : Γ → Γ is an isometry if for all γ, δ ∈ Γ we have

|γ -1 δ| S = | f (γ) -1 f (δ)| S .
The group of bijective isometries of a Cayley graph (Γ, S) is the group denoted by Iso(Γ, S) of all bijections f : Γ → Γ which are isometries. The action of Γ by left multiplication on itself yields a canonical injective group homomorphism Γ → Iso(Γ, S). Thus, Γ identifies naturally as a subgroup of Iso(Γ, S). Each time the generating set is clear in the context, we will write Iso(Γ) instead of Iso(Γ, S). In the literature, the group Iso(Γ) is often referred to as the automorphism group of the Cayley graph (Γ, S Γ ) and denoted by Aut(Γ, S Γ ).

Let Γ = S Γ and Λ = S Λ be two countable groups. A map f : Γ → Λ is an isometry if for all γ, δ ∈ Γ we have

|γ -1 δ| S Γ = | f (γ) -1 f (δ)| S Λ .
Let Iso((Γ, S Γ ), (Λ, S Λ )) be the space of bijective isometries between the Cayley a conull set. A bimeasurable bijection T : (X, µ) → (Y, ν) between two standard probability spaces is a p.m.p. isomorphism if T * µ = ν. A p.m.p. automorphism T ∈ Aut(X, µ) is aperiodic if the T-orbit of µ-almost every x ∈ X is infinite. A p.m.p. action of a countable group Γ on (X, µ) is a homomorphism α : Γ → Aut(X, µ). It is essentially free if for each γ ∈ Γ \ {1 Γ }, the p.m.p. automorphism α(γ) is aperiodic. If there is no need to give a name to the p.m.p. action α, we simply write γx instead of α(γ)x.

A p.m.p. partial automorphism on (X, µ) is a bimeasurable bijection T : A → B between measurable subsets A, B ⊆ X, which preserves the measure, that is, for all measurable subset C ⊆ A, one has µ(T(C)) = µ(C). A graphing on (X, µ) is a countable set Θ := {T i : A i → B i | i ∈ I} of p.m.p. partial automorphisms. The shortest path distance d Θ (x, y) between two distinct points x, y ∈ X is the smallest integer n ∈ N ∪ {+∞} such that there exists i 1 , . . . , i n ∈ I and ε 1 , . . . , ε n ∈ {±1} such that y = T ε 1 i 1 . . . T ε n i n (x). This defines an (extended) metric d Θ : X × X → N ∪ {+∞}.

Definition 5.2.2 (Isometric graphings). -Two graphings Θ on (X, µ) and Ξ on (Y, ν) are measurably isometric if there exists a p.m.p. isomorphism Φ : (X, µ) → (Y, ν) such that for µ-a.e. x, x ∈ X we have

d Ξ (Φ(x), Φ(x )) = d Θ (x, x ).
Definition 5.2.3 (Isometric orbit equivalence). -Let Γ = S Γ and Λ = S Λ be two countable groups. Two p.m.p. actions Γ α (X, µ) and Λ β (Y, ν) are isometric orbit equivalent if the graphings α(S Γ ) on (X, µ) and β(S Λ ) on (Y, ν) are measurably isometric.

In the definition of isometric orbit equivalence, both groups Γ and Λ come with their fixed generating system. It is thus noteworthy that the notion of isometric orbit equivalence heavily depends on the generating systems of the groups.

Example 5.2.4. -Let T ∈ Aut(X, µ) be an aperiodic transformation. Let C 2 be the cyclic group of order two, and ν be the uniform probability measure on C 2 . Let α be the p.m.p. Z-action on (X × C 2 , µ ⊗ ν) defined by

α(n)(x, ε) := (T n (x), ε) if n is even, (T n (x), 1 -ε) if n is odd.
Let β be the p.m.p. action of the infinite dihedral group D Then the graphings α({±1}) and β({a, b}) are measurably isometric. Thus, if Z and τ y ∈ [Λ, Γ] be defined as in Remark 5.2.8. Since Φ is length-preserving, we get by the cocycle identity that for all γ, δ ∈ Γ,

∞ := a, b | a 2 = b 2 = 1 on (X × C 2 , µ ⊗ u)
|σ x (γ) -1 σ x (δ)| S Λ = |σ(γ -1 δ, δ -1 x)| S Λ = |γ -1 δ| S Γ .
This means that σ x ∈ Iso 1 (Γ, Λ). Similarly, we have τ y ∈ Iso 1 (Γ, Λ). Moreover, the map x ∈ X → σ x ∈ Iso 1 (Γ, Λ) is Γ-equivariant and the map y ∈ Y → τ y ∈ Iso 1 (Λ, Γ) is Λ-equivariant. Thus, the pushforward of µ by x → σ x is a randisometry from Γ to Λ. Similarly, the pushforward of ν by y → τ y is a randisometry from Λ to Γ.

Lemma 5.2.11. -Let Γ = S Γ and Λ = S Λ be two countable groups. Two p.m.p. essentially free actions Γ α (X, µ) and Λ β (Y, ν) are isometric orbit equivalent if and only if there exists a length-preserving orbit equivalence between them.

Proof. If there exists a length-preserving orbit equivalence Φ between α and β, then we conclude by Lemma 5.2.6 that α and β are isometric orbit equivalent. Let us prove the converse. Assume that α and β are isometric orbit equivalent. Let Φ : (X, µ) → (Y, ν) be a pmp isomorphism such that for µ-almost every

x, x ∈ X, d α(S Γ ) (x, x ) = d β(S Λ ) (Φ(x), Φ(x )).
This implies that d α(S Γ ) (x, x ) is finite if and only if x and x are in the same Γorbit. Similarly, the distance d β(S Λ ) (Φ(x), Φ(x )) is finite if and only if Φ(x) and Φ(x ) are in the same Λ-orbit. Thus, we deduce that Φ is an orbit equivalence. Let σ : Γ × X → Λ and τ : Λ × Y → Γ be the orbit equivalence cocycles associated with Φ. By definition, for all γ ∈ Γ and µ-almost all x ∈ X, we have

d α(S Γ ) (x, α(γ)x) = d β(S Λ ) (Φ(x), β(σ(γ, x))Φ(x)).
Since the actions are essentially free, then the left hand side is equal to |γ| S Γ while the right hand side is equal to |σ(γ, x)| S Λ . We thus get that |σ(γ, x)| S Λ = |γ| S Γ . Similarly, we get that |τ(λ, y)| S Γ = |λ| S Λ for all λ ∈ Λ and ν-almost every y ∈ Y. This proves that Φ is a length-preserving orbit equivalence.

A canonical isometric orbit equivalence

Let Γ = S Γ and Λ = S Λ be two finitely generated groups. The space Iso(Γ, Λ) of bijective isometries between the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) is a locally compact, totally disconnected space when equipped with the topology of pointwise convergence. The space Iso(Γ) of bijective isometries of the Cayley graph (Γ, S Γ ) is a totally disconnected, locally compact group when endowed with the topology of pointwise convergence. Moreover, it contains naturally Γ as a lattice, and thus is unimodular. let σ x : Γ → Λ be defined by σ x (γ) := σ(γ -1 , x) -1 . By Remark 5.2.10, we obtain that σ x ∈ Iso 1 (Γ, Λ) for µ-almost every x ∈ X. Thus, there exists a bijective isometry between the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ).

Conversely, assume that the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric. Let µ and ν be the Haar probability measures on Iso 1 (Γ, Λ) and Iso 1 (Λ, Γ) respectively. By Corollary 5.3.2, the p.m.p. actions Γ (Iso 1 (Γ, Λ), m Γ ) and Λ (Iso 1 (Λ, Γ), ν) are isometric orbit equivalent. If these actions are essentially free, then the proof is complete. Else, we fix two p.m.p. essentially free actions Γ (X, µ X ) and Λ (Y, µ Y ) and consider the p.m.p. actions.

Γ (Iso 1 (Γ, Λ) × X × Y, µ ⊗ µ X ⊗ µ Y ), γ( f , x, y) := (γ • f , γx, f (γ -1 ) -1 y), Λ (Iso 1 (Λ, Γ) × X × Y, ν ⊗ µ X ⊗ µ Y ), λ( f , x, y) := (λ • f , f (λ -1 ) -1 x, λy).
These actions are essentially free. Moreover, as a direct consequence of Lemma 5.3.1, there is a length-preserving orbit equivalence between them, which implies by Lemma 5.2.11 that they are isometric orbit equivalent.

Remark 5.3.4. -The trick used at the end of the proof for getting essentially free actions while staying (isometric) orbit equivalent is due to Gaboriau [Gab02, Thm. 2.3].

Rigidity of isometric orbit equivalence

The aim of this section is to understand isometric orbit equivalence when the space of bijective isometries Iso(Γ, Λ) between Γ = S Γ and Λ = S Λ is countable. Observe that the cardinality of Iso(Γ), Iso(Λ) and Iso(Γ, Λ) coincide, because the groups Iso(Γ) and Iso(Λ) acts simply transitively on Iso(Γ, Λ).

We say that two countable groups Γ and Λ are virtually isomorphic if there exists finite index subgroups Γ 0 ≤ Γ and Λ 0 ≤ Λ which are isomorphic. We say that two p.m.p. actions Γ (X, µ) and Λ (Y, ν) are virtually measurably isomorphic if there exist finite index subgroups Γ 0 ≤ Γ and Λ 0 ≤ Λ, as well as a Γ 0 -invariant subset X 0 ⊆ X of positive measure and a Λ 0 -invariant subset Y 0 ⊆ Y of positive measure, such that the p.m.p. actions Γ 0 (X 0 , µ X 0 ) and Λ 0 (Y 0 , ν Y 0 ) are measurably isomorphic. We prove a rigidity result for isometric orbit equivalence. The strategy of the proof is modeled on the proof of orbit equivalence rigidity phenomena due to Furman [Fur99].

Theorem 5.4.1. -Let Γ = S Γ and Λ = S Λ be two finitely generated groups, such that the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric. Let Γ α (X, µ) and Λ β (Y, ν) be two p.m.p. essentially free actions that are isometric orbit equivalent. Assume that Iso(Γ), equivalently Iso(Λ), is countable. Then Γ and Λ are virtually isomorphic groups and the p.m.p. actions α and β are virtually measurably isomorphic. If in addition, every finite index subgroup of Γ acts ergodically on (X, µ), then Γ and Λ are isomorphic and α and β are measurably isomorphic.

Proof. Thanks to Lemma 5.2.11, we fix a length-preserving orbit equivalence Φ : (X, µ) → (Y, ν) between α and β. Let σ : X × Γ → Λ and τ : Λ × Y → Γ be the orbit equivalence cocycles associated with Φ. For x ∈ X and y ∈ Y, let σ x : Γ → Λ and τ y : Λ → Γ be the maps defined by σ x (γ) := σ(γ -1 , x) -1 and τ y (λ) := τ(λ -1 , y) -1 . By Remark 5.2.10, we know that σ x ∈ Iso 1 (Γ, Λ) and τ y ∈ Iso 1 (Λ, Γ) for µ-almost every x ∈ X and ν-almost every y ∈ Y. Moreover, the map x → σ x is Γ-invariant and the map y → τ y is Λ-invariant. Since Iso(Γ) is countable, the space Iso(Γ, Λ) is countable and thus the compact subset Iso 1 (Γ, Λ) is finite. Thus there is f 0 ∈ Iso 1 (Γ, Λ) such that the set X 0 := {x ∈ X : σ x = f 0 } satisfies µ(X 0 ) > 0. We define

Γ 0 := {γ ∈ Γ : γ • f 0 = f 0 }.
This is a finite index subgroup of Γ, which leaves X 0 invariant. Indeed, for all γ ∈ Γ 0 and x ∈ X 0 , we have σ α(γ)x = γ • σ x = γ • f 0 = f 0 . Let g 0 ∈ Iso 1 (Λ, Γ) be the inverse of f 0 and let Y 0 := {y ∈ Y : τ y = g 0 }.

We define Λ 0 := {λ ∈ Λ : λ • g 0 = g 0 }.

This is a finite index subgroup of Λ, which leaves Y 0 invariant. We know by Lemma 5.2.7, that for µ-almost every x ∈ X, for all γ ∈ Γ and λ ∈ Λ, τ(σ(γ, x), Φ(x)) = γ and σ(τ(λ, Φ(x)), x) = λ.

Thus, the maps σ x and τ Φ(x) are inverses of one another, that is σ x • τ Φ(x) = id Λ and τ Φ(x) • σ x = id Γ . Therefore, we have Φ(X 0 ) = Y 0 . Thus the map Φ induces an orbit equivalence (still denoted by) Φ : (X 0 , µ X 0 ) → (Y 0 , µ Y 0 ) between the p.m.p. actions Γ 0 (X 0 , µ X 0 ) and Λ 0 (Y 0 , ν Y 0 ). The orbit equivalence cocycles σ 0 : Γ 0 × X 0 → Y 0 and τ 0 : Λ 0 × Y 0 → Γ 0 associated with this orbit equivalence are independent of the space variable. Indeed for all γ ∈ Γ 0 , λ ∈ Λ 0 , σ 0 (γ, x) = σ(γ, x) = f 0 (γ -1 ) -1 for µ-almost every x ∈ X 0 , τ 0 (λ, y) = τ(λ, y) = g 0 (λ -1 ) -1 for ν-almost every y ∈ Y 0 . Thus, the groups Γ 0 and Λ 0 are isomorphic and the p.m.p. actions Γ 0 (X 0 , µ X 0 ) and Λ 0 (Y 0 , ν Y 0 ) are measurably isomorphic. This proves that the groups Γ and Λ are virtually isomorphic and that the actions α and β are virtually isomorphic.

If in addition, every finite index subgroup of Γ acts ergodically on (X, µ), then µ(X 0 ) = 1. Since Φ is a p.m.p. isomorphism, we deduce that ν(Y 0 ) = 1. In order to prove that α and β are measurably isomorphic, it remains to show that Γ 0 = Γ and Λ 0 = Λ. Up to null set, one can assume that X 0 is a Γ-invariant full measure set. Thus, for all γ ∈ Γ and x ∈ X 0 , we have γ • f 0 = γ • σ x = σ α(γ)x = f 0 . Thus Γ 0 = Γ. One proves similarly that Λ 0 = Λ. We therefore conclude that Γ and Λ are isomorphic and that the p.m.p. actions α and β are measurably isomorphic.

Weakly mixing actions are examples of p.m.p. actions for which every finite index subgroup acts ergodically. Concrete examples of weakly mixing actions are Bernoulli shifts. Therefore, we have the following result.

Corollary 5.4.2. -Let Γ = S Γ be a finitely generated group. Assume that Iso 1 (Γ) is finite. Let (A, κ) be a probability space. Any p.m.p. action Λ β (Y, ν) of some finitely generated group Λ = S Λ which is isometric orbit equivalent to the Bernoulli shift Γ (A, κ) Γ is actually measurably isomorphic to it and Λ is isomorphic to Γ.

Example 5.4.3. -Here are examples of finitely generated groups Γ = S such that Iso 1 (Γ, S) is finite. Leemann and de la Salle proved that any finitely generated group Γ admits a finite generating system S such that Iso 1 (Γ, S) is finite [START_REF] Leemann | Cayley graphs with few automorphisms: the case of infinite groups[END_REF]. For some finitely generated groups Γ, the set Iso 1 (Γ, S) is finite for all finite generating systems S. For instance, let Γ be a finitely generated, torsion free group which is either of polynomial growth, or a lattice in a simple Lie group G (in case G SL 2 (R), assume that Γ is uniform in G). Then for any finite generating system S of Γ, the space Iso 1 (Γ, S) is finite. These facts are due to Trofimov for groups with polynomial growth [START_REF] Trofimov | Graphs with polynomial growth[END_REF] and to Furman for lattice in simple Lie groups [START_REF] Furman | Mostow-Margulis rigidity with locally compact targets[END_REF]. We refer to [dlST19, Sec. 6] for a discussion about these results. Other examples of such groups are obtained by Guirardel and Horbez. They proved the following result: if Γ is a torsion-free finite index subgroup of the group of outer automorphisms of the free group F d on d ≥ 3 generators, then for any finite generating systems S of Γ, the space Iso 1 (Γ, S) is finite [START_REF] Guirardel | Measure equivalence rigidity of Out(F N )[END_REF]. Remark 5.4.4. -The result of Corollary 5.4.2 is false if Iso 1 (Γ, Λ) is infinite. For instance, let Λ 1 and Λ 2 be two non-isomorphic finite groups. Let Γ = S Γ be an infinite, finitely generated group and let Γ i := Λ i * Γ for i ∈ {1, 2}, equipped with the finite generating set Λ i ∪ S Γ . By a co-induction argument, one can show that the Bernoulli shifts Γ 1 ([0, 1], Leb) Γ 1 and Γ 2 ([0, 1], Leb) Γ 2 are isometric orbit equivalent, see for instance [Bow11, Thm. 1.1]. However, one can choose Λ 1 , Λ 2 and Γ so that Γ 1 and Γ 2 are not isomorphic. This question is related to the following problem. The measurable edge chromatic number of the graphing given by the Bernoulli shift F d ([0, 1], Leb) F d is ψ f ,q ∈ Iso(Γ) be the map defined by ψ f ,q (γ) := s(q) f (γ). We define the subset D ⊆ Iso(Γ) by D := {ψ f ,q : f ∈ Iso 1 (Γ), q ∈ Γ/Λ}.

Claim. -The set D is a fundamental domain for Iso(Γ)/Λ, that is, for all f ∈ Iso(Γ), there exists a unique λ ∈ Λ such that the map γ → λ -1 f (γ) belongs to D.

Proof of the claim. Let f ∈ Iso(Γ). Let δ = f (1 Γ ). Then there exists λ ∈ Λ and q ∈ Γ/Λ such that δ = λs(q). Observe that γ → δ -1 f (γ) belongs to Iso 1 (Γ). Then the map γ → λ -1 f (γ) belongs to D, because it coincides with ψ δ -1 f ,q .

claim This yields an action Iso(Γ) D, defined for all g ∈ Iso(Γ) and ψ ∈ D by (g, ψ) → (γ → λ -1 ψ(g -1 (γ))),

where λ is the unique element of Λ such that the map γ → λ -1 ψ(g -1 (γ)) belongs to D. If we denote by µ D the Haar measure on Iso(Γ) which satisfies µ D (D) = 1, then the action Iso(Γ) D preserves µ D and is measurably isomorphic to the p.m.p. action Iso(Γ) (Iso(Γ)/Λ, m Λ ). Let us prove that the action Iso(Γ) (D, µ D ) is measurably isomorphic to the action Iso(Γ)

(Iso 1 (Γ) × Γ/Λ, µ ⊗ u) defined for g ∈ Iso(Γ) and ( f , q) ∈ Iso 1 (Γ) × Γ/Λ by g( f , q) := (g • f , f (g -1 (1 Γ )) -1 q).

Let us define a map Φ : Iso 1 (Γ) × Γ/Λ → D by the formula Φ( f , q) := γ → ψ f ,q (γ).

This is a bijection, as it is a surjective map by definition of D and it is straightforward to check that it is an injective map. Moreover, by definition of µ and µ D , we get Φ * (µ ⊗ u) = µ D . It is a straightforward computation to check that Φ intertwines the actions Iso(Γ) Iso 1 (Γ) × Γ/Λ and Iso(Γ) D, which finishes the proof of the lemma. Theorem 5.5.2. -Let Γ = S Γ be a finitely generated group. Let Λ ≤ Γ be a finite index subgroup. Let µ Γ be the Haar probability measure on Iso(Γ)/Γ and u be the uniform probability measure on Γ/Λ. Then the p.m.p. action Γ (Iso(Γ)/Λ, m Λ ) is isometric orbit equivalent to the diagonal action Γ (Iso(Γ)/Γ × Γ/Λ, m Γ ⊗ u).

Proof. In this proof, we will denote by * the p.m.p. action Γ (Iso(Γ)/Λ, m Λ ) and by the diagonal action Γ (Iso(Γ)/Γ × Γ/Λ, m Γ ⊗ u). Let u be the uniform probability measure on Γ/Λ. We know by Lemma 5.5.1 that * is measurably isomorphic to the action Γ (Iso 1 (Γ) × Γ/Λ, µ ⊗ u), still denoted by * and given by γ * ( f , q) := (γ • f , f (γ -1 ) -1 q). two in Iso(F d ). Thus, up to conull set, we have sA = (Iso 1 (F d ) × F d /Λ) \ A. Therefore sU = V. By Lemma 5.5.6, we obtain that Λ is not contained in F ev d , which concludes the proof. Corollary 5.5.8. -Let Λ ≤ F d be a finite index subgroup which is not included in the even subgroup F ev d . Then the p.m.p. action F d (Iso(F d )/Λ, m Λ ) and the diagonal action F d (Iso(F d )/F d × F d /Λ, m F d ⊗ u) are isometric orbit equivalent but the former is mixing, whereas the latter is not.

A concrete isometric orbit equivalence for F 2

We finish this section with a concrete example of two p.m.p. ergodic actions of F 2 that are isometric orbit equivalent but not measurably isomorphic.

Let F 2 be the free group on two generators a and b and let S = {a ±1 , b ±1 } be the standard generating system. Let |•| S be the word length associated with S. The Cayley graph (F 2 , S) is isomorphic to the 4-regular tree. Let Iso(F 2 ) be the group of bijective isometries of (F 2 , S), that is, the group of all bijections f : F 2 → F 2 such that for all γ, δ ∈ F 2 ,

| f (γ) -1 f (δ)| S = |γ -1 δ| S .
Let C be the space of proper colorings with five colors on the vertex set of the Cayley graph (F 2 , S). That is, an element of C is a map col : F 2 → {1, 2, 3, 4, 5} such that for all γ ∈ F 2 and for all s ∈ S, we have col(γ) = col(γs). The set C is a closed, thus compact, subspace of {1, 2, 3, 4, 5} F 2 . The group Iso(F 2 ) acts on C and we denote by * the action, which is defined as follows: for all f ∈ Iso(F 2 ) and c ∈ C, the coloring f * col is given by γ → col( f -1 (γ)). This action is simply transitive and it admits a unique invariant probability measure µ, which can be constructed as follows. First, choose uniformly at random the color of the identity element 1 F 2 . Then, by moving radially outwards 1 F 2 in the Cayley graph, extend the coloring at each vertex by choosing the color uniformly at random among the admissible ones, independently at each vertex. Thus we get a p.m.p. action * of F 2 on (C, µ). Let us explain briefly why this action is mixing. Fix two distinct 5-cycles A, B ∈ Sym({1, . . . , 5}) such that for all i ∈ {1, . . . , 5}, A(i) = B(i). For instance, take A := (1 2 3 4 5) and B := (1 3 5 2 4). This yields a transitive action F 2 {1, 2, 3, 4, 5} by letting the generator a act like A and b act like B. Let Λ be the stabilizer of the point 1. If m Λ denotes the Haar probability measure on the quotient Iso(F 2 )/Λ, then it can be proved that the p.m.p. action * of F 2 on (C 2 , µ) is measurably isomorphic to F 2 (Iso(F 2 )/Λ, m Λ ). By definition, F 2 {1, . . . , 5} is isomorphic to the action F 2 F 2 /Λ. As there is no bipartition of {1, . . . , 5} whose pieces are exchanged by any element s ∈ S, we get by Lemma 5.5.6 and Theorem 5.5.7 that the p.m.p. action * of F 2 on (C, µ) is mixing.

Let us construct another p.m.p. action of F 2 on (C, µ), that we denote by , which is isometric orbit equivalent to * , but which is not mixing. We define by the action of the generators a and b of F 2 . For col ∈ C, we let a col := s * col, where s is the unique element in {a ±1 , b ±1 } such that (s * col)(1 F 2 ) = A(col(1 F 2 )). Such an element s exists because c is a proper vertex coloring. Similarly, we define b col := t * col, where t is the unique element in {a ±1 , b ±1 } such that (t * col)(1 F 2 ) = B(col(1 F 2 )). This defines a p.m.p. action of F 2 on (C, µ). Observe that for each i ∈ {1, . . . , 5}, the set {col ∈ C : col(1 Γ ) = i} is invariant by Λ. Thus, the p.m.p. action is not mixing. Moreover, by construction, the actions * and are isometric orbit equivalent. This yields a concrete illustration of Theorem 5.5.2. Indeed, it can be showed that:

-the p.m.p. action * is measurably isomorphic to F 2 (Iso(F 2 )/Λ, m Λ ), where m Λ is the Haar probability measure on Iso(F 2 )/Λ, -the p.m.p. action is measurably isomorphic to the diagonal action F 2 (Iso(F 2 )/F 2 × F 2 /Λ, m F 2 ⊗ u) where m F 2 is the Haar probability measure on Iso(F 2 )/F 2 and u is the uniform probability measure on F 2 /Λ. Résumé : Cette thèse se situe à l'interface entre dynamique topologique et dynamique mesurée. Premièrement, j'y étudie la notion d'action allostérique. Ce sont des actions génériquement libres au sens topologique mais pas génériquement libres au sens de la mesure. Ce comportement étonnant met en valeur les nuances entre sous-groupes aléatoires invariants et sous-groupes uniformément récurrents. Un second sujet d'étude est l'équivalence orbitale quantitative, qui renforce l'équivalence orbitale. Il s'agit de comprendre comment les structures métriques sur les orbites des actions peuvent être distordues par équivalence orbitale. Une grande partie des travaux de cette thèse gravite autour d'un des théorèmes fondateurs de cette théorie : le théorème de Belinskaya.

Mots clés : Dynamique topologique, dynamique mesurée, sous-groupes aléatoires invariants, allostérie, équivalence orbitale quantitative, théorème de Belinskaya.

Abstract : This PhD thesis lies at the interface between topological dynamics and measurable dynamics. First, I study the notion of allosteric actions. These actions are generically free in the sense of the topology but not generically free in the sense of the measure. This surprising behavior highlights the differences between invariant random subgroups and uniformly recurrent subgroups. The nascent theory of quantitative orbit equivalence is the second topic of this thesis. This is a strengthening of orbit equivalence, which aims to understand how metric structures on the orbits of the actions can be distorted. A large part of my work gravitates around one of the founding result of this theory: Belinskaya's theorem.

Keywords : Topological dynamics, measurable dynamics, invariant random subgroups, allostery, quantitative orbit equivalence, Belinskaya's theorem

  Introduction en français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 :

 2 Belinskaya's theorem is optimal . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3: Cycles in ϕ-integrable full groups . . . . . . . . . . . . . . . . . . . . . . . . 119 Chapter 4: Quantitative orbit equivalence between Z and D ∞ . . . . . . . . . . . . . . . . 129 Part III Quantitative orbit equivalence and graphings 141 Chapter 5: Isometric orbit equivalence for probability-measure preserving actions . . . . . 143 Index 171

Partie III : Équivalence orbitale quantitative et graphages Chapitre 5 :

 5 Équivalence orbitale isométrique pour les actions préservant une mesure de probabilité, article prépublié arXiv:2203.14598

  Preuve. Soit Z (C, µ) une action minimale ergodique qui n'est pas essentiellement libre. Par définition µ({x ∈ C : Stab(x) = {0}}) > 0.

  µ) une action minimale ergodique. L'application mesurable Stab : x → {γ ∈ Γ : γ • x = x} associe à tout point de C un sous-groupe de Γ. Par ailleurs Stab est équivariante puisque pour tout x ∈ C et γ ∈ Γ, on a Stab(γ • x) = γStab(x)γ -1 .

Notons a et b les

  deux générateurs de F 2 . Les permutations A et B définissent une action du groupe libre F 2 sur C p,n en posant a • x := A(x) et b • x := B(x) pour tout x ∈ C p,n . Cette action est illustrée dans la Figure 1.1. Appelons bord de C p,n l'ensemble ∂C p,n des points de profondeur 2n. Ce sont exactement les points fixes de a. Par ailleurs, un calcul élémentaire donne |∂C p,n | |C p,n | -→ n→+∞ p -2 p -1 .

Figure 1 .

 1 Figure 1.1. -L'ensemble C 3,1 et les permutations A (en noir) et B (en rouge).

d

  Γ (x, y) := inf{|γ| Γ : γ • x = y}. Cela fournit une L'un des objectifs principaux de l'équivalence orbitale quantitative est de comprendre comment une équivalence orbitale entre deux actions p.m.p. peut distordre les structures métriques associées. Plus précisément, soit Γ et Λ deux groupes de type fini et soit |•| Γ et |•| Λ les longueurs des mots associées à un système fini de générateurs de Γ et Λ respectivement. Définition. -Soit p ∈]0, +∞]. Une équivalence orbitale Φ :

Figure 1

 1 Figure 1.4. -Une portion d'une involution non-croisante.

Figure 1

 1 Figure 1.5. -Les demi-cercles associés à P et Q.

  Continuum of allosteric actions for non-amenable surface groups prepublished article arXiv:2110.01068 Part II: Quantitative orbit equivalence for Z-actions Chapter 2: Belinskaya's theorem is optimal prepublished article with A. Carderi, F. Le Maître and R. Tessera, arXiv:2201.06662 Chapter 3: Cycles in ϕ-integrable full groups Chapter 4: Quantitative orbit equivalence between Z and D ∞ Part III: Quantitative orbit equivalence and graphings Chapter 5: Isometric orbit equivalence for probability-measure preserving actions, prepublished article arXiv:2203.14598

Figure 2

 2 Figure 2.1. -The set C 3,1 and the permutations A in black, B in red.

Figure 2

 2 Figure 2.4. -A portion of a non-crossing involution.

Figure 2

 2 Figure 2.5. -The circular arcs associated with P and Q

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.2.1 Topological dynamic and URS/IRS . . . . . . . . . . . . 58 1.2.2 Profinite actions and their URS/IRS . . . . . . . . . . . . 59 1.2.3 Allostery and commensurability . . . . . . . . . . . . . . 61 1.3 Finite index subgroups of surface groups . . . . . . . . . . . . 63 1.3.1 Residual properties of surface groups . . . . . . . . . . . 63 1.3.2 Special kind of finite index subgroups in surface groups 64 1.4 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . 66

Figure 1

 1 Figure 1.1. -An illustration of the morphism that erases generators.

Figure 1

 1 Figure 1.2. -Illustrations of the proof of Theorem 1.3.4. The above line illustrates the coverings corresponding to the inclusions Λ ≤ Λ d ≤ Γ. The bottom line illustrates the covering corresponding to the inclusion N ≤ Λ d /N.

  4.1. Remark 1.4.4. -Let α : Γ (C, µ) be an allosteric action. Then we have supp(IRS(α)) ⊂ {Stab α (x) | x ∈ C}.

Figure 1

 1 Figure 1.3. -A branched surface

  By definition, we have supp(T A ) = A and x, y ∈ A are in the same T-orbit if and only if they are in the same T A -orbit. Whenever T is aperiodic, the first return time n T,A coincides with the T-cocycle c T A of T A . Lemma 2.2.3. -Let T ∈ Aut(X, µ), let P ∈ Aut(X, µ) be a periodic transformation and D a fundamental domain of P. Let U := T D P. Then the following are true. (i) U D = T D . (ii) If x ∈ D and n(x) is the cardinality of the P-orbit of x, then n U,D (x) = n(x).

  Definition 2.3.19. -The dynamical entropy of a measure-preserving transformation T ∈ Aut(X, µ) is the infimum of the Shannon entropies of its Tdynamically generating functions.The above definition is not the standard definition, however it is equivalent by a theorem of Rokhlin [Rok67, Thm. 10.8]. Also note that by definition, the dynamical entropy of T ∈ Aut(X, µ) is finite if and only if T admits a dynamically generating function of finite entropy.Proposition 2.3.20. -Let T ∈ Aut(X, µ) be an aperiodic transformation with infinite dynamical entropy and let U ∈ [T] be a transformation whose T-cocycle has finite Shannon entropy. Then U has infinite dynamical entropy.

  Theorem 2.4.1. -The metric d ϕ,T is complete, separable and right-invariant on [T] ϕ and the topology generated by d ϕ,T is a group topology. In particular [T] ϕ is a Polish group. Proof. We have already shown in Lemma 2.2.14 that d ϕ,T is right-invariant. Corollary 2.2.18 tells us that the inclusion [T] ϕ → [T] is Lipschitz and in particular any d ϕ,T -Cauchy sequence is d u -Cauchy. Since d u is complete, any d ϕ,T -Cauchy sequence has a d u -limit. Claim. -Let (U n ) n≥0 be a d ϕ,T -Cauchy sequence of elements of [T] ϕ and let

. 5 )

 5 Corollary 2.2.18 tells us that [T] ϕ and [T] ψ embed continuously in [T]. Therefore Lemma 2.4.3 yields that the inclusion map of [T] ψ into [T] ϕ is continuous.

  ) and thus also in ([T] ϕ , d ϕ,T ). Moreover, we have {U ∈ APER ∩ [T] ϕ : T and U have the same orbits} = ε∈Q * + n≥1 O ε,n , which is a countable intersection of open sets, thus by definition a G δ set.

  4.12 yields that U Y k tends to U while Corollary 2.3.7 gives us that T X\Y k tends to the identity. Claim 2. -For k large enough, we have thatV k ∈ O F ,ε .Proof of the claim. For all k ≥ 0, putF k := {A ∩ Y k : A ∈ F }. Since U Y k is weakly mixing on Y k , we have U Y k ∈ O F k ,ε/5. By construction, the transformations U Y k and V k coincide on Y k , so we also have V k ∈ O F k ,ε/5 . Since lim k µ(Y k ) = 1, for k large enough and all A ∈ F , we have µ(A (A ∩ Y k )) < ε/5. We thus get that V k ∈ O F ,ε for k large enough by Claim 1. claim It follows immediately from Claim 2 that any ergodic element in [T] ϕ is a limit of aperiodic elements in O F ,ε ∩ [T] ϕ . This shows the inclusion (2.8), ending the proof of the theorem.

  Theorem 2.A.3 (Belinskaya's theorem). -Let T ∈ Aut(X, µ) be ergodic, let U ∈ [T] 1 . Then T and U are flip-conjugate.

  Now, observe that if A and B are two sets, then1 A B = |1 A -1 B |. Thus, we are left to prove that the cardinal of the set [[1, n]] [[c(x) + 1, c(x) + n]] is equal to 2 min(|c(x)|, n). There are two cases to check. If n ≤ |c(x)|, then the intervals [[1, n]] and [[c(x) + 1, c(x) + n]] are disjoint, thus the cardinal of their symmetric difference is 2n = 2 min(|c(x)|, n). If |c(x)| < n, then the cardinal of the intersection of [[1, n]] and [[c(x) + 1, c(x) + n]] is n -|c(x)| and the cardinal of their union is n + |c(x)|. Thus, the cardinal of their symmetric difference is 2|c(x)| = 2 min(|c(x)|, n), which finishes the proof.

Y

  +∞ := {y ∈ Y : τ(ξ ab , y) = +∞} and Y -∞ := {y ∈ Y : τ(ξ ab , y) = -∞} are both invariant under the action of ab. Moreover, if τ(ξ ab , y) = +∞, then τ(ξ ba , y) = -∞. This implies that β(a)Y +∞ = Y -∞ . Thus, the measures of Y a and of Y b are non null. This shows that the subgroup of D ∞ generated by ab doesn't act ergodically. We construct a new p.m.p. action of Z on (Y, ν) that we Proof. Let z ∈ [[x, y]]. Since P([[x, y]]) = [[x, y]], we get that P(z) ∈ [[x, y]] and thus [[z, P(z)]] ⊆ [[x, y]] because P is non-crossing. This implies that for all z ∈ [[x, y]],

Figure 4

 4 Figure 4.2. -The P upper arcs and the TPT -1 lower arcs

  defined by the action of the two generators β(a)(x, 0) := (T(x), 1), β(a)(x, 1) := (T -1 (x), 0), β(b)(x, 1) := (T(x), 0), β(b)(x, 0) := (T -1 (x), 1).

  Question 5.4.5. -Let F d be the free group on d ≥ 2 generators x 1 , . . . , x d and let S Γ := {x ±1 1 , . . . , x ±1 d }. Let Λ = S Λ be a finitely generated group. Consider a p.m.p. essentially free action Λ (Y, ν) which is isometric orbit equivalent to the Bernoulli shift F d ([0, 1], Leb) F d . Does this imply that Λ is isomorphic to F d and that the p.m.p. actions are measurably isomorphic?

Figure 5

 5 Figure 5.1. -On the left, black arrows represent the permutation A, red arrows B. On the right, the portion of an element c ∈ C, for which a col = b * col, a -1 col = a -1 * col, b col = b -1 * col and b -1 col = a * col.

Table des matières de l'introduction en français Partie

  Minimalité et ergodicité . . . . . . . . . . . . . . . . . . . Notions de liberté . . . . . . . . . . . . . . . . . . . . . . Allostérie . . . . . . . . . . . . . . . . . . . . . . . . . . . Exemples de groupes non-allostériques . . . . . . . . . . Sous-groupes aléatoires invariants / Sous-groupes uniforméments récurrents . . . . . . . . . . . . . .

I : Allostérie . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1 , et même L ∞ , s'avère être riche pour les actions de Z d lorsque d ≥ 2. Par exemple, Fieldsteel et Friedman démontrent que pour tout d ≥ 2 et toute action ergodique Z d α (X, µ), il existe une action mélangeante Z d β (Y, ν) telle que α et β sont L ∞ orbitalement équivalentes. Nous démontrons qu'un tel résultat est faux en équivalence orbitale isométrique. En fait, nous démontrons un résultat de rigidité du même type pour les groupes Γ munis d'un système fini de générateurs S Γ pour lesquels le groupe d'automorphismes du graphe de Cayley (Γ, S Γ ) est dénombrable. Un théorème de Trofimov permet de démontrer que le groupe d'automorphismes du graphe de Cayley de Z d , muni de n'importe quel système fini de générateurs S, est en effet dénombrable, voir [MS98, Thm. 4.3]. L'archétype d'un groupe qui admet un graphe de Cayley dont le groupe d'automorphismes est non-dénombrable est le groupe libre F d à d ≥ 2 générateurs. Nous démontrons que le phénomène de rigidité obtenu pour Z d ne s'applique pas pour F d . Rostislav I. Grigorchuk. Some topics in the dynamics of group actions on rooted trees. Proc. Steklov Inst. Math., 273 :64-175, 2011.

	[Gri11]
	[GW15]	Eli Glasner and Benjamin Weiss. Uniformly recurrent subgroups.
		In Recent trends in ergodic theory and dynamical systems. International
		conference in honor of S. G. Dani's 65th birthday, Vadodara, India, De-
		cember 26-29, 2012. Proceedings, pages 63-75. Providence, RI : Ame-
		rican Mathematical Society (AMS), 2015.
	[Jos21]	Matthieu Joseph. Continuum of allosteric actions for non-amenable
		surface groups, 2021.
	[LBMB18] Adrien Le Boudec and Nicolás Matte Bon. Subgroup dynamics and
		C * -simplicity of groups of homeomorphisms. Ann. Sci. Éc. Norm.
		Supér. (4), 51(3) :557-602, 2018.

Théorème. -Soit d ≥ 2 et soit S un système de générateurs fini pour Z d . Soit Z d α (X, µ) une action p.m.p. mélangeante. Toute action Z d (Y, ν) qui est isométriquement orbitalement équivalente à α lui est conjuguée. Théorème. -Soit F d le groupe libre à d ≥ 2 générateurs x 1 , . . . , x d et soit S le système fini de générateurs S := {x ±1 1 , . . . , x ±1 d }. Alors il existe des actions p.m.p. isométriquement orbitalement équivalentes F d α (X, µ) et F d β (Y, ν) telles que α est mélangeante mais β ne l'est pas. [Lig02] Thomas M. Liggett. Tagged particle distributions or how to choose a head at random. In In and out of equilibrium. Probability with a physics flavor. Papers from the 4th Brazilian school of probability, Mambucaba, Brazil, August 14-19, 2000, pages 133-162. 2002.

Table of contents of the introduction in English Part

  Invariant random subgroups / Uniformly recurrent subgroups . . . . . . . . . . . . . . . . . . . . . . . . Examples of allosteric groups . . . . . . . . . . . . . . . . A construction of allosteric actions for F 2 . . . . . . . . .

I : Allostery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimality and ergodicity . . . . . . . . . . . . . . . . . . Notions of freeness . . . . . . . . . . . . . . . . . . . . . . Allostery . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of non-allosteric groups . . . . . . . . . . . . . Some allosteric/non-allosteric groups . . . . . . . . . . . Part II: Quantitative orbit equivalence for Z actions . . . . . . . . . Orbit equivalence . . . . . . . . . . . . . . . . . . . . . . . Example of an orbit equivalence . . . . . . . . . . . . . . Orbit equivalence and invariants . . . . . . . . . . . . . . Quantitative orbit equivalence . . . . . . . . . . . . . . . Example of an L p orbit equivalence . . . . . . . . . . . . Return of invariants . . . . . . . . . . . . . . . . . . . . . Belinskaya's theorem is optimal . . . . . . . . . . . . . . Non-crossing involution . . . . . . . . . . . . . . . . . . . Quantitative orbit equivalence between Z and D ∞ . . . Part III: Quantitative orbit equivalence and graphings . . . . . . . Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . Isometric orbit equivalence . . . . . . . . . . . . . . . . . Rigidity and flexibility . . . . . . . . . . . . . . . . . . . .

uniformly recurrent subgroup (URS) is

  a closed, minimal, Γ-invariant subset of Sub(Γ). Let Γ (C, µ) be a minimal ergodic action. The URS associated with this action is the unique URS contained in the closure of {Stab(x) : x ∈ C}, see[GW15]. One can show that this URS is contained in the support of the IRS associated with the action [Jos21, Lem. 2.2]. Moreover, this URS is equal to {{1 Γ }} if and only if the action is topologically free [LBMB18, Prop. 2.7]. We thus get the following characterization of allostery. Lemma. -A minimal ergodic action Γ (C, µ) is allosteric if and only if the trivial group {1 Γ } is strictly contained in the support of the IRS associated with this action.

	Thus, the pushforward of µ by Stab yields a Γ-invariant ergodic probability
	measure on Sub(Γ), that is, an ergodic IRS. By definition, the action is essentially
	free if and only if this IRS is equal to the measure δ {1 Γ } .
	Invariant random subgroups have a topological counterpart. A

  Proposition 1.2.8. -Let (Γ n ) n≥0 be a chain in Γ, and let α be the corresponding ergodic profinite Γ-action. If URS(α) is trivial, then either IRS(α) is trivial, or there exists a finite index subgroup Λ ≤ Γ such that the p.m.p. Λ-action by conjugation on (Sub(Γ), IRS(α)) is not ergodic.Proof. Assume that the p.m.p. Γ-action by conjugation on (Sub(Γ), IRS(α)) remains ergodic under any finite index subgroup of Γ. Since URS(α) is trivial, there exists by Lemma 1.2.7 a sequence (γ n ) n≥0 of elements in Γ such that

	n≥0

  So far, every left coset x ∈ Γ/Λ d is fixed by every element of Γ A , and either every or none of the left coset x ∈ Γ/Λ d is fixed by δ, depending on whether δ ∈ Λ d or not.

	that the following diagram commutes
	Γ B
	( (

  3: The group Λ d /N is a residually finite p-group. We let π : Λ d Λ d /N be the quotient group homomorphism. Since Λ d /N is an orientable surface group, it is a residually finite p-group by Theorem 1.3.2. Thus, there exists a normal subgroup N Λ d /N whose index is a power of p, such that for every k ∈ {0, . . . , d -1} \ E, for every i ∈ A, π(a i,k ) / ∈ N and π(α i,k ) / ∈ N . If γ ∈ Λ d , we also assume that π(γ) / ∈ N, and if δ ∈ Λ d , we also assume that for all k

  We are now ready to prove Theorem 1.4.1 and Theorem 1.4.2.Proof of Theorem 1.4.1. Let Γ be a non-amenable orientable surface group, and we fix a decomposition Γ = Γ A * Z Γ B . Let 0 < t < 1 be a real number. Let (p n ) n≥1 be a sequence of pairwise distinct prime numbers. We fix a sequence (r n ) n≥1 such that each r n belongs to ]0, 1[∩Z[1/p n ] and ∏ n≥1 r n = t. Such a sequence exists because each Z[1/p n ] is dense in R. Finally, let (γ n ) n≥0 be an enumeration of the elements in Γ with γ 0 = 1, and (δ n ) n≥1 be an enumeration of the elements inΓ B \ Z Γ B .For every n ≥ 1, there exists by Theorem 1.3.4 a finite index subgroup Λ t n ≤ Γ which doesn't contain γ n , whose index [Γ : Λ t n ] is a power of p n , such that the number of left cosets x ∈ Γ/Λ t n that are fixed by any element of Γ A is equal to r n [Γ : Λ t n ], and such that none of the left coset x ∈ Γ/Λ t n is fixed by δ n . For every n ≥ 1, let Γ t n:= Λ t 1 ∩ • • • ∩ Λ t n .The sequence (Γ t n ) n≥1 forms a chain in Γ and we denote by α t the corresponding ergodic profinite action, and by µ t the profinite Γ-invariant probability measure on lim ← -Γ/Γ t n . This is a p.m.p. ergodic minimal action and we will prove that it is allosteric. By construction of Λ t n , we have that

an isomorphism, and it is Γ-equivariant.

n≥1 Γ t n = {1 Γ }.

This implies by Lemma 1.2.7 that URS(α t ) is trivial. Let us prove that each element of Γ B \ Z Γ B acts essentially freely for α t . Let δ ∈ Γ B \ Z Γ B . By Lemma 1.4.3, the number of x ∈ Γ/Γ t n such that δx = x is equal to the number of

Table of contents of Chapter 2

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quantitative orbit equivalence and full groups . . . . . . . . . 2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 ϕ-integrable orbit equivalence and full groups . . . . . . 2.2.3 Metric properties of ϕ-integrable full groups . . . . . . . 2.3 Flexibility of ϕ-integrable orbit equivalence . . . . . . . . . . . 2.3.1 Construction of cycles in ϕ-integrable full groups . . . . 2.3.2 Construction of ϕ-integrable orbit equivalences . . . . . 2.3.3 Connection to Shannon orbit equivalence . . . . . . . . . 2.3.4 Finiteness of entropy and Shannon orbit equivalence . . 2.

4 Weakly mixing elements are generic in

[T] ϕ . . . . . . . . . . . 2.4.1 Polish group topology . . . . . . . . . . . . . . . . . . . . 2.4.2 A sublinear ergodic theorem for ϕ-integrable functions . 2.4.3 Continuity properties of the first return map . . . . . . . 2.4.4 Optimality of Belinskaya's theorem . . . . . . . . . . . . 2.4.5 Weakly mixing elements form a dense G δ set . . . . . . . 2.A Proof of Belinskaya's theorem . . . . . . . . . . . . . . . . . . .

  2.3.2. -The hypothesis that ϕ is sublinear is necessary, as the result is false for L 1 full groups of certain aperiodic transformations. Indeed if T ∈ Aut(X, µ) is ergodic, then there exists an n-cycle in [T] 1 whose support is A ⊆ X if and only if exp(2iπ/n) is in the spectrum of the restriction of T A to A [LM18, Thm. 4.8]. In particular the L 1 full group of the Bernoulli shift contains no ncycle with full support for any n ≥ 2. By contrast, Theorem 2.3.1 says that as soon as p < 1, its L p full group contains an n-cycle of full support for every n ≥ 2.

Example 2.3.3. -In certain concrete situations, we can exhibit explicit involutions. Let T be the Bernoulli shift on ({0, 1}, κ)

  The inequality . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.1.2 Shift-coupling inequality and ϕ-integrability . . . . . . . 121 Fundamental domains of cycles . . . . . . . . . . . . . . 123 3.2.2 Cycles in the full group of a Bernoulli shift . . . . . . . . 124

	Table of contents of Chapter 3
	3.1 Shift-coupling inequality . . . . . . . . . . . . . . . . . . . . . . 119
	3.1.1 3.2 Application to ϕ-integrable full groups . . . . . . . . . . . . . 122
	3.2.1

  Lemma 3.1.4. -Let ϕ : R + → R + be a metric-compatible function. Then for all 0 < s ≤ t, we have ϕ(t)/t ≤ 2ϕ(s)/s.Proof. Let n = t/s . Since s ≤ t, we have n ≥ 1. Since t < (n + 1)s, we get that ϕ(t) ≤ ϕ((n + 1)s) ≤ (n + 1)ϕ(s). Finally, we divide by t and use the inequality ns ≤ t to obtain

	ϕ(t) t	≤	n + 1 t	ϕ(s) ≤	n + 1 n	ϕ(s) s	≤ 2	ϕ(s) s	.
	Definition 3.1.5. -Let ϕ : R + → R + be a metric-compatible function. A
	measurable map f : X

ἄλλος : autre, στερεός : exprime une idée de fixité.

Muni de la topologie produit, l'espace C := ∏ k≥0 C p k ,n k est compact. On le munit de la mesure de probabilité µ, produit des mesures de probabilité uniformes sur chaque C p k ,n k .Proposition. -L'action diagonale de F 2 sur C est allostérique. Esquisse de preuve. Tout d'abord F 2 (C, µ) est une action minimale ergodique. En effet, une application du lemme des restes chinois permet de montrer que pour tout K ≥ 0, l'action diagonale F 2

Xd Z (x, T impair (x)) p dµ.Comme il suffit de s'assurer de l'intégrabilité sur les générateurs des groupes Z et Z 2 , on en déduit que les actions Z (X, µ) et Z 2 (X, µ) sont L <1/2 orbitalement équivalentes.Puisque ces actions sont libres et orbitalement équivalentes, on peut associer à presque tout point x ∈ X une applicationf x : Z → Z 2 où f x (n) est l'unique élément (u, v) ∈ Z 2 tel que T(x) = (T pair ) u (T impair ) v (x). Lorsque l'on regarde précisément le graphe de f x pour un x ∈ X générique, on reconnaît la courbe de Lebesgue, voir Figure1.3. Cette condition "p < 1/2" n'est pas sans rappeler le fait qu'une courbe continue [0, 1] → [0, 1] 2 qui remplit le carré ne peut pas être p-Hölder pour p > 1/2. En partitionnant N en fonction du reste modulo n, m ≥ 2, on obtient d'une façon similaire deux actions p.m.p. Z n (X, µ) et Z m (X, µ) qui sont L <p avec p = min(n/m, m/n). On peut retrouver ces exemples (sous une forme moins explicite) dans l'article [DKLMT20, Thm. 6.9]. Nous expliquons ci-dessous en quoi ces conditions sur p sont optimales.

ἄλλος: other, στερεός: a notion of fixity.

ἄλλος: other, στερεός: fix, firm, solid, rigid

Step 2: Erasing the right amount of generators. Let n be the length of γ ∈ Γ \ {1 Γ } in the generating set S. In the sequel we let d be a (large enough) power of the prime p such that rd is an integer, and rd + n ≤ d. Let E ⊂ {n + 1, . . . , d -1 -n} be a subset of cardinality rd, so that γ doesn't belong to the normal closure N of the set ∪ k∈E b k j 0 Γ A b -k j 0 in Λ d . Let us prove that none of the conjugate of δ by a power of b j 0 belongs to N. Assume this is not the case, then this would imply that δ belongs to the normal closure of ∪ d-1 k=0 b k j 0 Γ A b -k j 0 in Λ d , which is easily seen to be equal to the normal closure Γ A Γ of Γ A in Γ.But the group Γ/ Γ A Γ is naturally isomorphic to Γ B / Z Γ B , in such a way

We can actually characterize when [T] ϕ = [T] ψ and more generally when [T] ϕ ≤ [T] ψ , see Proposition

2.4.2.

We actually only need Step 1 from the simpler proof given in[START_REF] Eigen | Multiple Rokhlin tower theorem: A simple proof[END_REF].

Remerciements

Chapter 2

In the context of ϕ-orbit equivalence, it is natural to consider the the set of measure-preserving transformations U whose cocycle c U is ϕ-integrable. In order for this set to be a group, the following conditions on ϕ are required. Definition 2.2.7. -A function ϕ : R + → R + is metric-compatible if:

• (subadditivity) for all s, t ∈ R + , ϕ(s + t) ≤ ϕ(s) + ϕ(t).

• (separation) ϕ(0) = 0 and ϕ(t) > 0 for all t > 0.

• (monotonicity) ϕ is a non-decreasing function.

Example 2.2.8. -Any concave function ϕ : R + → R + that satisfies ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 is metric-compatible. In particular for every p ≤ 1, the function ϕ(t) = t p is metric-compatible. It is moreover sublinear whenever p < 1. Other examples of sublinear metric-compatible functions are given by ϕ(t) = log(1 + t) or ϕ(t) = t/ log(2 + t).

The term "metric-compatible" was coined because of the following property: whenever d is a metric on a set X, then ϕ • d is also a metric on X.

Convention. -For all t ∈ R, we use the notation

The map (s, t) → |s -t| ϕ is a metric on R. Definition 2.2.9. -Let ϕ : R + → R + be a metric-compatible function. The ϕ-integrable full group of an aperiodic transformation T ∈ Aut(X, µ) is

Given a metric-compatible function ϕ, the ϕ-integrable full group [T] ϕ is indeed a group: given U, V ∈ [T] ϕ , the cocycle identity implies that

We then get that

(2.4)

Example 2.2.10. -If ϕ is any metric-compatible function which is bounded, then [T] ϕ = [T] and if ϕ is the identity map, then we recover the L 1 full group [T] 1 defined by the third named author in [LM18]. Any other such ϕ gives rise

We can now recall a continuity property satisfied by the first return map in the full group, which was first observed by Keane.

Lemma 2.4.9 ([Kea70, Lem. 3]). -Let T be a measure-preserving transformation, then the map

It is worth noting that the analogue of Lemma 2.4.8 fails for the L 1 -full group. Indeed, let T ∈ Aut(X, µ) be ergodic and let ϕ := id R + Then Kac's Lemma yields that for all measurable A ⊆ X of positive measure, d ϕ,T (T A , id) = d ϕ,T (T, id) = 1. Since T ∅ = id, this shows that the map MAlg(X, µ) → [T] 1 defined by A → T A is not continuous.

However, the situation is not that clear when ϕ is sublinear.

Question 2.4.10. -Let ϕ : R + → R + be a sublinear metric-compatible function.

Is the map MAlg(X, µ) → [T] ϕ defined by A → T A continuous? More generally, is the map [T] ϕ × MAlg(X, µ) → [T] ϕ given by (U, A) → U A continuous?

In this section we give two partial answers to the above questions. We first prove that the map A → U A satisfies a continuity property "from below". For this, we need the following version of Scheffé's lemma for sequences of Z-valued ϕ-integrable functions.

Lemma 2.4.11. -Let f : X → Z be a measurable function and let ( f n ) n≥0 be a sequence of measurable functions f n : X → Z that converge in measure to f . If

Proof. It suffices to show that given ε > 0, there is δ > 0 such that for all measurable function g : X → Z satisfying

(2.6) one has that X | f -g| ϕ dµ ≤ ε. To this end, fix ε > 0. Since X | f | ϕ dµ < +∞, by Lebesgue's dominated convergence theorem there exists δ 0 > 0 such that for all measurable subset A ⊆ X, if µ(A) < δ 0 then A | f | ϕ dµ < ε. Take δ := min{δ 0 , ε}. Let g : X → Z be a measurable function satisfying (2.6). If we let A := {x ∈ X : f (x) = g(x)}, we have

This is in fact a group, which is one of the main objects of study of [CJLMT22].

Fundamental domains of cycles

The support of a measure-preserving transformation T ∈ Aut(X, µ) is the measurable set supp(T) := {x ∈ X : T(x) = x}. A measure-preserving transformation P ∈ Aut(X, µ) is periodic if the P-orbit of almost every x ∈ X is finite. A fundamental domain of a periodic transformation P ∈ Aut(X, µ) is a measurable subset A ⊆ X which intersects almost every P-orbit at exactly one point. Let m ≥ 2. A measure-preserving transformation P ∈ Aut(X, µ) is a m-cycle if for almost every x ∈ X, the cardinality of the P-orbit of x is either 1 or m. Here we prove the following result.

Theorem 3.2.2. -Let ϕ : R + → R + be a sublinear metric-compatible function. Let T ∈ Aut(X, µ) be an aperiodic transformation. Fix an integer m ≥ 2 and let P ∈ [T] ϕ be a m-cycle with full support. For any fundamental domain D ⊆ X of P, we have

Proof. Since P is a m-cycle with full support, the sets D, P(D), . . . , P m-1 (D) form a partition of X. For all k ∈ {0, . . . , m -1} and x ∈ P k (D), let c(x) := c P -k (x), where we recall that c P -k denotes the T-cocycle of

Proof. Let A ⊆ X be measurable subset. We know that D, P(D), . . . ,

Finally, D is a fundamental domain of a m-cycle with full support, thus µ(D) = 1/m, which concludes the proof of the claim.

123

Chapter 3

We apply the claim to get that for all k ≥ 1, the measure (T k+c ) * µ is absolutely continuous with respect to (T k ) * µ and its Radon-Nikodym derivative is given by

Since T preserves the measure µ, we have

Finally, Lemma 3.1.1 yields

To conclude, we observe that c : X → Z is ϕ-integrable. Indeed, on each piece P k (D) of the finite partition D, P(D), . . . , P m-1 (D), the map c coincides with c P -k , which is ϕ-integrable since P -k ∈ [T] ϕ . Proposition 3.1.6 then yields to the desired conclusion.

Cycles in the full group of a Bernoulli shift

In this section, we apply Theorem 3.2.2 in the concrete case of Bernoulli shifts.

Theorem 3.2.3. -Let ϕ be a metric-compatible function. Let A be a finite space, κ the uniform measure on A, and m := |A|. Let T ∈ Aut(A Z , κ Z ) be the Bernoulli shift. Let P ∈ [T] ϕ be a m-cycle with full support. If there exists a ∈ A such that {(x n ) n∈Z ∈ A Z : x 0 = a} is a fundamental domain for P, then

Proof. Let X := A Z and µ := κ Z . Let a ∈ A such that the set D := {(x n ) n∈Z ∈ X : x 0 = a} is a fundamental domain for P. For all k ≥ 0, we let X k be the random variable defined by X k (x

is a sequence of i.i.d. random variables. For all n ≥ 0, let S n := X 1 + . . . X n . By the central limit theorem, there is a constant C > 0 such that

Chapter 4

Quantitative orbit equivalence between Z and D ∞ In this chapter, we focus on quantitative aspects of orbit equivalences between p.m.p. essentially free actions of two particular groups: the group of integers Z and the infinite dihedral group D ∞ , which admits the presentation

These groups are both infinite and amenable, so the theorem of Ornstein and Weiss [OW80] implies that any p.m.p. ergodic action of Z is orbit equivalent to any p.m.p. ergodic action of D ∞ . We prove the following quantitative result. Theorem 4.0.1. -Let Z α (X, µ) be an ergodic action. Then the following statements holds.

(i) There exists an ergodic action D ∞ β (Y, ν) such that α and β are L <1 orbit equivalent.

(ii) By contrast, the action α is L 1 orbit equivalent to some ergodic action of D ∞ if and only if the subgroup 2Z doesn't act ergodically on (X, µ).

We refer to Section 4.1 for the definitions of L 1 and L <1 orbit equivalence. We prove item (ii) of Theorem 4.0.1 in Section 4.2. For this we use a result due to Bowen, which says that any L 1 orbit equivalence extends to the end compactifications of the groups [START_REF] Bowen | Integrable orbit equivalence rigidity for free groups[END_REF]. In Section 4.3, we prove a combinatorial result on a special kind of involutions defined on Z that we call non-crossing. In Section

Proof. Since 2Z does not act ergodically, there is a partition X = A B into measurable sets of measure 1/2 that are both invariant under 2Z. We define a p.m.p. action D ∞ β (X, µ) by the actions of its generators a and b. For all x ∈ X, we let

By construction, the actions α and β are L ∞ orbit equivalent (they are even isometric orbit equivalent), which concludes the proof.

Non-crossing involutions

Definition and first properties

In this section, we denote by T the addition by 1 in Z, that is T(x) := x + 1 for all x ∈ Z. For all k, l ∈ Z, we denote by [[k, l]

Remark 4.3.2. -This is equivalent to saying that for all x, y ∈ Z, the intervals [[x, P(x)]] and [[y, P(y)]] are either disjoint or one is contained in the other. The non-crossing property has the following pictorial interpretation. For all x ∈ Z such that P(x) = x, we draw the circular arc in the upper half-plane whose extremities are x and P(x), which is perpendicular to the horizontal axis. Then the involution P is non-crossing if and only if none of these circular arcs intersect. We refer to these arcs as the P upper arcs in the sequel. We will focus on fixed-point free involutions P : Z → Z, that is, satisfying P(x) = x for all x ∈ Z. We explain below some properties of fixed-point free, non-crossing involutions that will be useful in the sequel. Lemma 4.3.3. -Let P : Z → Z be a fixed-point free, non-crossing involution. Then TPT -1 is a fixed-point free, non-crossing involution.

Proof. The map TPT -1 is an involution. Let us prove that it is non-crossing. For all x ∈ Z, we have

This proves that TPT -1 is non-crossing. Finally TPT -1 (x) = x if and only if PT -1 (x) = T -1 (x). Since P has no fixed point, this proves that TPT -1 has no fixed point. Lemma 4.3.4. -Let P : Z → Z be a fixed-point free, non-crossing involution. Then for all x ∈ Z, the number P(x)x is odd.

. This implies that P induces an involution of the interval [[x, P(x)]]. This implies that the cardinal of [[x, P(x)]] is even, because P has no fixed point. In other words, P(x)x is odd. Lemma 4.3.5. -Let P : Z → Z be a fixed-point free, non-crossing involution. Let x, y ∈ Z such that x < y and P

Chapter 4

The L <1 full group of T, denoted by [T] <1 , is the group of all U ∈ [T] such that for all p < 1,

Let x, y ∈ X be two points in the same T-orbit. Then there is a unique n ∈ Z such that T n (x) = y. We denote by [[x, y]] T the interval with extremities x and y, which is defined by

The following result is an easy consequence of Rokhlin's lemma [Rok67].

Lemma 4.4.1. -Let T ∈ Aut(X, µ) be ergodic. Then for all l ≥ 1, there exists B ⊆ X of positive measure which is l-separated for T.

The measure of such a set B satisfies µ(B) ≤ 1/l. We will use l-separated sets to construct involutions in full groups which are non-crossing.

Let T ∈ Aut(X, µ) be ergodic and let A ⊆ X be of positive measure. The first return time of T to A is the map n T,A : A → N ∪ {+∞} defined for all x ∈ A by

If we denote by µ A the measure on A induced by µ, that is, the measure defined for all B ⊆ A by µ A (B)

We now prove that there exist non-crossing involutions with full support in L <1 full groups. This result is implicitly contained in the proof of [CJLMT22, Thm. 3.1]. We provide here a complete proof. Theorem 4.4.3. -Let T ∈ Aut(X, µ) be ergodic. Then there exists a non-crossing involution P ∈ [T] <1 with full support.

Proof. Let T 1 := T and A 1 = X. By Lemma 4.4.1, we fix an 2-separated subset B 1 ⊆ A 1 of positive measure. For all x ∈ A 1 , let n 1 (x) be the smallest integer n ≥ 0 such that x ∈ T n 1 (B 1 ). This is finite for µ-almost every x ∈ X by the Poincaré recurrence theorem. Let A 2 ⊆ A 1 be the measurable subset given by

If µ(A 2 ) > 0, then we construct n 2 . We denote by T 2 : A 2 → A 2 the first return map of T 1 with respect to A 2 . By Lemma 4.4.1, one gets B 2 ⊆ A 2 a 2 2separated subset for T 2 of positive measure. Then for all x ∈ A 2 , we define n 2 (x) as the smallest integer n ≥ 0 such that x ∈ T n 2 (B 2 ). Let A 3 ⊆ A 2 be the measurable subset given by

If µ(A 3 ) > 0, we next construct n 3 . We let T 3 be the first return map of T 2 with respect to A 3 and we consider B 3 ⊆ A 3 a 2 3 -separating subset for T 3 of positive measure. Then for all x ∈ A 3 we define n 3 (x) as the smallest integer n ≥ 0 such that x ∈ T n 3 (B 3 ).

Assume that this construction can be run indefinitely. We then obtain a sequence of decreasing subsets of positive measure X = A 1 ⊇ A 2 ⊇ . . . , a sequence of measure preserving transformations T k : A k → A k , a sequence B k ⊆ A k of 2 k -separated sets for T k of positive measure, and a sequence of maps n k : A k → N. For all k ≥ 1, the map T k+1 is the first return time of T k on A k+1 , where

This implies that the sequence of sets (A k \ A k+1 ) k≥1 forms a partition of X. We now define the non-crossing involution P. For all k ≥ 1 and all x ∈ A k \ A k+1 , we let

By construction, P is a non-crossing involution with full support contained in the full group [T]. We need to prove that P ∈ [T] <1 . Let c P be the T-cocycle of P. For all x ∈ A 1 \ A 2 , we have |c

Chapter 5

Isometric orbit equivalence for probability-measure preserving actions

The content of this chapter is the same as that of the article [Jos22]

Introduction

The purpose of this chapter is to compare probability-measure preserving actions of finitely generated groups via the graphings they define. A graphing on a probability space (X, µ) is a graph G whose vertex set is X and whose edge set is a measurable, symmetric subset of X × X, such that the equivalence relation R "belonging to the same connected component of G" satisfies the following two conditions: (i) each R-class is countable, (ii) any bimeasurable bijection T : A → B between measurable subsets A, B ⊆ X, such that (x, T(x)) ∈ R for all x ∈ X, preserves the measure µ.

Graphings have seen recently a sharp rise in interest as they play an important role in graph limit theory. Indeed, they serve as limit objects for sequences graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric. Let Γ α (X, µ) and Λ β (Y, ν) be two p.m.p. essentially free actions that are isometric orbit equivalent. If Iso(Γ) (equivalently Iso(Λ)) is countable, then (i) There exists finite index subgroups Γ 0 ≤ Γ and Λ 0 ≤ Λ which are isomorphic.

(ii) There exists a Γ 0 -invariant subset X 0 ≤ X of positive measure and a Λ 0 -invariant subset Y 0 ≤ Y of positive measure such that the p.m.p. actions Γ 0 (X 0 , µ X 0 ) and Λ 0 (Y 0 , ν Y 0 ) are measurably isomorphic.

If in addition, every finite index subgroup of Γ acts ergodically on (X, µ), then Γ and Λ are isomorphic and α and β are measurably isomorphic.

We Therefore, mixing is not invariant under isometric orbit equivalence. Theorem 5.1.4 is false for F 1 Z. In fact, for ergodic actions of Z, Belinskaya proved in [Bel68] that any two p.m.p. ergodic actions of Z that are L 1 orbit equivalent are flip-conjugate, that is, measurably isomorphic up to an automorphism of the group Z.

Theorem 5.1.4 is proved using a general construction of isometric orbit equivalent actions that we explain now. For a concrete illustration of Theorem 5.1.4, we refer to Section 5.5.3.

Let Γ be a finitely generated group and fix S Γ a finite generating system. Let Λ ≤ Γ be a finite index subgroup. The groups Λ and Γ both act by postcomposition on Iso(Γ). Then we prove in Section 5.5.1 that the quotient action Γ Iso(Γ)/Λ, equipped with its Haar probability measure, is isometric orbit Chapter 5 graphs (Γ, S Γ ) and (Λ, S Λ ). We say that the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric if Iso((Γ, S Γ ), (Λ, S Λ )) is nonempty. Again, if the generating systems are clear in the context, we will write Iso(Γ, Λ) instead of Iso((Γ, S Γ ), (Λ, S Λ )).

Example 5.2.1. -Here are some examples of finitely generated groups with isometric Cayley graphs.

1. Let Γ = Z be equipped with its usual generating system S Γ = {-1, +1}.

Let Λ be the infinite dihedral group D ∞ , that is, the free product of two copies of the cyclic group C 2 of order two. A presentation of D ∞ is given by

Let S Λ be the usual generating system {a, b}. Then the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric.

2. More generally, let Γ = F d be the free group on d ≥ 2 generators x 1 , . . . , x d , and let S Γ be the generating system {x ±1 1 , . . . , x ±1 d }. Let Λ be the free product of 2d copies of C 2 , whose presentation is given by

and S Λ be the usual generating system {a 1 , . . . , a 2d }. Then the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric.

3. Let m, n ≥ 2 be two integers. Let F m and F n be the free group on m generators x 1 , . . . , x m and on n generators y 1 , . . . , y n respectively. Let Γ m,n be the direct product F m × F n and S Γ m,n be the generating system

It follows from the work of Burger and Mozes [START_REF] Burger | Groups acting on trees: From local to global structure[END_REF][START_REF] Burger | Lattices in product of trees[END_REF] that for appropriate values of m and n, the Cayley graph (Γ m,n , S Γ m,n ) is isometric to the Cayley graph of some virtually simple group.

Dyubina observed in [Dyu00

] that there exist finitely generated groups Γ = S Γ and Λ = S Λ with isometric Cayley graphs, such that Γ is solvable but Λ not virtually solvable. This illustrates the fact that several algebraic properties are not preserved under the property of having isometric Cayley graphs.

Definition of isometric orbit equivalence

Let (X, µ) be a standard probability space. A bimeasurable bijection T : X → X is a p.m.p. automorphism of (X, µ) if for all measurable set A ⊆ X, one has µ(T -1 (A)) = µ(A). We denote by Aut(X, µ) the group of all p.m.p. automorphisms of (X, µ), two such automorphisms being identified if they coincide on Chapter 5

is equipped with the generating system S Z := {±1} and D ∞ with the generating system S D ∞ , then the p.m.p. actions α and β are isometric orbit equivalent.

Example 5.2.5. -Let Γ = S Γ be a finitely generated group. Let Γ α (X, µ) be a p.m.p. action and define

Fix a proper d-coloring on E, that is, a measurable map c : E → {1, . . . , d} such that for µ-almost every x, x , x ∈ X, if (x, x ) and (x, x ) are distinct elements of E, then c(x, x ) = c(x, x ). This always exists when d is large enough (see [START_REF] Csóka | Kőnig's line coloring and Vizing's theorems for graphings[END_REF] for a precise statement) and the smallest such d is called the measurable edge chromatic number of the graphing α(S Γ ). Let Λ be the group given by the presentation

1 and let S Λ be the generating system {a 1 , . . . , a d }. Let β be the p.m.p. action of Λ defined by the action of its generators

Then the p.m.p. actions α and β are isometric orbit equivalent.

We now explain a way to prove that p.m.p. actions are isometric orbit equivalent. For this, we need to introduce the notion of length-preserving cocycle. Let Γ and Λ be two countable groups. Let Γ (X, µ) be a p.m.p. action. A measurable function σ : Γ × X → Λ is a cocycle if for all γ, δ ∈ Γ, σ(γδ, x) = σ(γ, δx)σ(δ, x) for µ-almost every x ∈ X.

Lemma 5.2.6. -Let Γ = S Γ and Λ = S Λ be two countable groups. Let Γ α (X, µ) and Λ β (Y, ν) be two p.m.p. actions. Assume that there is a p.m.p. isomorphism Φ : (X, µ) → (Y, ν) and two length-preserving cocycles σ :

Then α and β are isometric orbit equivalent.

Proof. Let us prove that for µ-almost every x, x ∈ X, we have

The reverse inequality is proved in a similar way, using the fact that τ is lengthpreserving. We thus obtain that

which proves that α and β are isometric orbit equivalent.

Isometric orbit equivalence for essentially free actions

Two p.m.p. actions of two countable groups Γ (X, µ) and Λ (Y, ν) are orbit equivalent if there exists an orbit equivalence between them, that is, a p.m.p. isomorphism Φ : (X, µ) → (Y, ν) such that for µ-almost every x ∈ X,

Consider now two p.m.p. essentially free actions Γ (X, µ) and Λ (Y, ν) and let Φ : (X, µ) → (Y, ν) be an orbit equivalence between them. We therefore have two maps, which are uniquely defined by freeness of the actions:

They are called the orbit equivalence cocycles associated with Φ. Moreover, they satisfy the following properties. (iii) (Fixing the identity) We have σ(1 Γ , x) = 1 Λ and τ(1 Λ , y) = 1 Γ .

Chapter 5

Remark 5.2.8. -Let [Γ, Λ] be the set of maps f :

A randomorphism from Γ to Λ is a Γ-invariant probability measure on [Γ, Λ], see [Mon06, Def. 5.2]. Monod observed that randomorphisms supported on bijections can be obtained via orbit equivalences as follows. Let Φ be an orbit equivalence between p.m.p. essentially free actions Γ (X, µ) and Λ (Y, ν). Let σ : Γ × X → Λ and τ : Λ × Y → Γ be the orbit equivalence cocycles associated with Φ. For x ∈ X and y ∈ Y, let σ x : Γ → Λ and τ y : Λ → Γ be defined by

We deduce by Lemma 5.2.7 that σ x is a bijection such that σ x (1 Γ ) = 1 Λ . Similarly, we get that τ y is a bijection such that τ y (1 Λ ) = 1 Γ . Moreover, the cocycle identity for σ and τ implies that

Thus, the pushforward of µ by x → σ x is a randomorphism from Γ to Λ supported on bijections. Similarly, the pushforward of ν by y → τ y is a randomorphism from Λ to Γ supported on bijections. Isometric orbit equivalent actions was defined in Definition 5.2.3 thanks to the graphings associated with the actions. We now characterize isometric orbit equivalent actions in terms of cocycles. In Lemma 5.2.11, we will prove that two essentially free actions are isometric orbit equivalent if and only if there exists an orbit equivalence whose associated cocycles are length-preserving. Definition 5.2.9. -Let Γ = S Γ and Λ = S Λ be two countable groups. Let Φ : (X, µ) → (Y, ν) be an orbit equivalence between two p.m.p. essentially free actions Γ (X, µ) and Λ (Y, ν). We say that Φ is a length-preserving orbit equivalence if the orbit equivalence cocycles σ : Γ × X → Λ and τ : Λ × X → Γ associated with Φ are length-preserving. Remark 5.2.10. -Let Γ = S Γ and Λ = S Λ be two countable groups. Let Iso 1 (Γ, Λ) be the set of

Inspired by the language of randomorphism proposed by Monod [START_REF] Monod | An invitation to bounded cohomology[END_REF], we say that a Γ-invariant probability measure on Iso 1 (Γ, Λ) is a randisometry from Γ to Λ. Randisometries can be obtained via length-preserving orbit equivalence as follows. Let Φ be a length-preserving orbit equivalence between two p.m.p. essentially free actions Γ (X, µ) and Λ (Y, ν). Let σ : Γ × X → Λ and τ : Λ × Y → Γ be the orbit equivalence cocycles associated with Φ.

Assume that Iso(Γ, Λ) is nonempty. Then the group Iso(Γ) acts simply transitively on Iso(Γ, Λ) by precomposition by the inverse. The group Iso(Λ) also acts simply transitively on Iso(Γ, Λ) by postcomposition. This implies that there exists a unique measure (up to a multiplicative constant) on Iso(Γ, Λ) which is invariant by the actions Iso(Γ) Iso(Γ, Λ) and Iso(Λ) Iso(Γ, Λ). We call it the Haar measure on Iso(Γ, Λ). The inverse map Iso(Γ, Λ) → Iso(Λ, Γ) is a bimeasurable bijection, which sends the Haar measure to the Haar measure. The pushforward of the Haar measure on Iso(Γ, Λ) by the quotient map Iso(Γ, Λ) → Iso(Γ, Λ)/Λ, rescaled to have mass 1, is called the Haar probability measure on Iso(Γ, Λ)/Λ. Let Iso 1 (Γ, Λ) be the compact open subspace of Iso(Γ, Λ) defined by

This is a fundamental domain for the action Λ Iso(Γ, Λ). Let m be the Haar measure on Iso(Γ, Λ) such that m(Iso 1 (Γ, Λ)) = 1. The restriction of m to Iso 1 (Γ, Λ) is called the Haar probability measure on Iso 1 (Γ, Λ). Let m Λ be the Haar probability measure on Iso(Γ, Λ)/Λ. We then obtain that the p.m.p. action Iso(Γ, Λ) (Iso(Γ, Λ)/Λ, m Λ ) is measurably isomorphic to the p.m.p. action Iso(Γ, Λ)

(Iso 1 (Γ, Λ), m) defined by

The restriction of this action to Γ ≤ Iso(Γ, Λ) boils down to the p.m.p. action Γ (Iso 1 (Γ, Λ), m) encountered in Remark 5.2.10 and given by

Let m Γ denote the Haar probability measure on Iso(Λ, Γ)/Γ. One of the aim of this section is to prove that the actions Γ (Iso(Γ, Λ)/Λ, m Λ ) and Λ (Iso(Λ, Γ)/Γ, m Γ ) are isometric orbit equivalent. In order to prove this, we will work with the Γ-action on Iso 1 (Γ, Λ) and the Λ-action on Iso 1 (Λ, Γ) instead. Lemma 5.3.1. -Let Γ = S Γ and Λ = S Λ be two finitely generated groups, such that the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric. Let µ be the Haar probability measure on Iso 1 (Γ, Λ) and ν the Haar probability measure on Iso 1 (Λ, Γ). Then the following are true.

(i) The map σ : Γ × Iso 1 (Γ, Λ) → Λ defined by σ(γ, f ) := f (γ -1 ) -1 is a lengthpreserving cocycle.

(ii) The map τ : Λ × Iso 1 (Λ, Γ) → Γ defined by τ(λ, f ) := f (λ -1 ) -1 is a lengthpreserving cocycle.

(iii) The inverse map Φ : (Iso 1 (Γ, Λ), µ) → (Iso 1 (Λ, Γ), ν) is a p.m.p. isomorphism, such that for all γ ∈ Γ, λ ∈ Λ, we have

Proof. We start by proving (i). The fact that σ is a cocycle is a straightforward computation:

This proves that σ is length-preserving. The proof of (ii) is identical. For the proof of (iii), it is clear that Φ is a bimeasurable map. Moreover, since the inverse map sends Haar measure to Haar measure, we obtain that Φ * µ = ν. Finally, the two formulas left to prove are straightforward computations.

We obtain the following result as a corollary.

Corollary 5.3.2. -Let Γ = S Γ and Λ = S Λ be two finitely generated groups, such that the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric. Let m Λ be the Haar probability measure on Iso(Γ, Λ)/Λ and m Γ the Haar probability measure on Iso(Λ, Γ)/Γ. Then the p.m.p. actions Γ (Iso(Γ, Λ)/Λ, m Λ ) and Λ (Iso(Λ, Γ)/Γ, m Γ ) are isometric orbit equivalent.

Proof. Let σ : Γ × Iso 1 (Γ, Λ) → Λ be defined by σ(γ, f ) := f (γ -1 ) -1 and τ : Λ × Iso 1 (Λ, Γ) → Γ be defined by τ(λ, f ) := f (λ -1 ) -1 . By Lemma 5.3.1, these are length-preserving cocycles. Let µ and ν be the Haar probability measures on Iso 1 (Γ, Λ) and Iso 1 (Λ, Γ) respectively. Then by Lemma 5.2.6 we get that the actions Γ (Iso 1 (Γ, Λ), µ) and Λ (Iso 1 (Λ, Γ), ν) are isometric orbit equivalent. Thus, the p.m.p. actions Γ (Iso(Γ, Λ)/Λ, m Λ ) and Λ (Iso(Λ, Γ)/Γ, m Γ ) are isometric orbit equivalent.

In general, the action Γ (Iso(Γ, Λ)/Λ, m Λ ) is not essentially free. However, a standard trick can be used to obtain p.m.p. essentially free actions that are isometric orbit equivalent.

Theorem 5.3.3. -Let Γ = S Γ and Λ = S Λ be two finitely generated groups. Then Γ and Λ admit p.m.p. essentially free actions that are isometric orbit equivalent if and only if the Cayley graphs (Γ, S Γ ) and (Λ, S Λ ) are isometric.

Proof. Assume that Γ (X, µ) and Λ (Y, ν) are p.m.p. essentially free actions that are isometric orbit equivalent. By Lemma 5.2.11, there exists a lengthpreserving orbit equivalence Φ : (X, µ) → (Y, ν). Let σ : Γ × X → Λ and τ : Λ × Y → Γ be the orbit equivalence cocycles associated with Φ. For x ∈ X, known to be either 2d or 2d + 1 [START_REF] Csóka | Kőnig's line coloring and Vizing's theorems for graphings[END_REF]. However, its exact value is unknown [START_REF] Kechris | Descriptive graph combinatorics[END_REF]Prob. 5 This is a consequence of the fact that there is a notion of entropy which distinguishes Bernoulli shifts up to measure-isomorphism and which is preserved under bounded orbit equivalence. For amenable groups, Kolmogorov-Sinai entropy is preserved under bounded orbit equivalence [START_REF] Austin | Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence[END_REF], whereas for free groups, the f -invariant is preserved under bounded orbit equivalence [START_REF] Bowen | Entropy for actions of free groups under bounded orbit-equivalence[END_REF].

Construction of isometric orbit equivalent actions

The general construction

Given a finitely generated group Γ = S Γ and a finite index subgroup Λ ≤ Γ, we explain in this section a construction of p.m.p. isometric orbit equivalent actions of Γ. In the sequel, we will use the following notation. For all g ∈ Iso(Γ) and all f ∈ Iso 1 (Γ), we denote by g • f the element of Iso 1 (Γ) defined by

We explained in Section 5.3 why the action (g, f ) → g • f is measurably isomorphic to the action Iso(Γ) Iso(Γ)/Γ. Beware here, that Iso(Γ)/Γ means the quotient of Iso(Γ) under the Γ-action on Iso(Γ) by postcomposition. Lemma 5.5.1. -Let Γ = S Γ be a finitely generated group. Let Λ ≤ Γ be a finite index subgroup. Let m Λ be the Haar probability measure on Iso(Γ)/Λ. Let µ be the Haar probability measure on Iso 1 (Γ) and u be the uniform probability measure on Γ/Λ. Then the p.m.p. action Iso(Γ) (Iso(Γ)/Λ, m Λ ) is measurably isomorphic to the p.m.p. action Iso(Γ)

(Iso

Proof. We fix a section s : Γ/Λ → Γ. For all f ∈ Iso 1 (Γ) and q ∈ Γ/Λ, we let Let Φ : Iso 1 (Γ) → Iso 1 (Γ) be the inverse map, defined by f

This is a p.m.p. isomorphism by Lemma 5.3.1. Morever, we have

By Lemma 5.3.1, the map (γ, f ) → f (γ -1 ) -1 is a length-preserving cocycle, thus we obtain by Lemma 5.2.6 that the p.m.p. actions * and are isometric orbit equivalent, which concludes the proof.

The case of the free group

In this section, we characterize the subgroups Λ ≤ F d for which the p.m.p. action F d (Iso(F d )/Λ, m Λ ) is mixing. Here, m Λ denotes the Haar probability measure. Before this, we need to give some properties of mixing actions of locally compact groups.

Let G be a locally compact, non-compact, second countable group. A function f : G → C vanishes at infinity if for all ε > 0, the set {g ∈

This group satisfies the Howe-Moore property. Theorem 5.5.5 (Lubotzky-Mozes [LM92], Pemantle [START_REF] Pemantle | Automorphism invariant measures on trees[END_REF]). -Any p.m.p. ergodic action of Iso ev (F d ) on a standard probability space is mixing.

Chapter 5

The following lemma characterizes the finite index subgroups of the even subgroup F ev d .

Lemma 5.5.6. -Let Λ ≤ F d be a finite index subgroup. Then the following are equivalent.

(i) Λ is not contained in F ev d .

(ii) For all γ ∈ F d , the index [F ev d : γΛγ (iv) There is no bipartition

We obtain (ii) by dividing both sides of the equality by [F d : F ev d ]. We now prove (ii)⇒(iii). Observe that for all γ ∈ F d , the group γΛγ -1 ∩ F ev Let us prove the contrapositive of (iii)⇒(iv). Assume that there exists a partition F d /Λ = U V such that for all s ∈ S, sU = V. Then we also have sV = U for all s ∈ S. By induction, we get that γU = U and γV = V for all γ ∈ F ev d . Thus, the action F ev d F d /Λ is not transitive. Finally, let us prove the contrapositive of (iv)⇒(i). Assume that Λ ≤ F ev (iii) Λ is not contained in the even subgroup F ev d .

Proof. In this proof, we denote by µ the Haar probability measure on Iso 1 (F d ).

Let • be the p.m.p. action Iso(F d ) (Iso 1 (F d ), µ) given by

If u denotes the uniform probability measure on F d /Λ, then by Lemma 5.5.1, the p.m.p. action Iso(F d ) (Iso(F d )/Λ, m Λ ) is measurably isomorphic to the p.m.p. action Iso(F d ) (Iso 1 (F d ) × F d /Λ, µ ⊗ u) given by g( f , q) := (g • f , f (g -1 (1 Γ )) -1 q).

The proof of (i)⇒(ii) is a direct consequence of Lemma 5.5.3. Let us prove (ii)⇒(iii). We prove the contrapositive. Assume that Λ is contained in F ev d . Then by Lemma 5.5.6, there is a bipartition F d /Λ = U V such that for all s ∈ S, we have sU = V. By induction, we get that γU = U and γV = V for all γ ∈ F ev d . For all f ∈ Iso 1 (F d ) and all γ ∈ F d , we have | f (γ -1 ) -1 | S = |γ| S . Thus we obtain that the sets Iso 1 (F d ) × U and Iso 1 (F d ) × V are invariant by F ev d . Thus, the p.m.p. action F d (Iso 1 (F d ) × F d /Λ, µ ⊗ u) is not mixing, which is equivalent to saying that F d (Iso(F d )/Λ, m Λ ) is not mixing. We now prove (iii)⇒(i). We prove the contrapositive. Assume that Iso(F d ) (Iso(F d )/Λ, m Λ ) is not mixing. That is, the action Iso(F d ) (Iso 1 (F d ) × F d /Λ, µ ⊗ u) is not mixing. By Lemma 5.5.4, the p.m.p. action Iso ev (F d ) (Iso 1 (F d ) × F d /Λ, µ ⊗ u) is not mixing and thus not ergodic by Theorem 5.5.5. Let A ⊆ Iso 1 (F d ) × F d /Λ be a measurable subset of measure 1/2 which is Iso ev (F d )invariant. We define the following set U := {q ∈ F 2 /Λ : µ ⊗ u(A ∩ (Iso 1 (F d ) × {q}) > 0}.

Observe that the subgroup Iso 1 (F d ) is contained in Iso ev (F d ). Moreover, for all g ∈ Iso 1 (F d ) and for all ( f , q) ∈ Iso 1 (F d ) × F d /Λ, we have g( f , q) = ( f • g -1 , q). Thus, the group Iso 1 (F d ) acts transitively on each Iso 1 (F d ) × {q}. Since A is Iso ev (F d )-invariant, we obtain that A = Iso 1 (F d ) × U up to a conull set. We claim that the set U and its complement V form a partition of F d /Λ such that for all s ∈ S, sU = V. Indeed, for s ∈ S, the facts that s / ∈ Iso ev (F d ) and that Iso ev (F d ) is normal in Iso(Γ) imply that sA is Iso ev (F d )-invariant. But sA cannot be equal to A, because otherwise A would be a Iso(F d )-invariant set of measure 1/2, contradicting the ergodicity of Iso(F d ) (Iso(F d )/Λ, m Λ ). Moreover, the intersection sA ∩ A cannot be of positive measure, otherwise it would be a Iso ev (F d )invariant set of measure < 1/2, which is impossible since Iso ev (F d ) has index Index