Il y a à peu près 10 ans j'ai commencé à faire des mathématiques ; remercier tous ceux que j'ai rencontrés, ceux qui m'ont aidé en écrivant des lettres de recommandation, ceux que j'ai écouté faire des exposés motivants, ceux qui m'ont raconté leurs idées, ceux qui ont fait un petit commentaire crucial sur ma recherche ou posé la question à laquelle j'aurais dû penser avant, me semble aujourd'hui impossible. Je sais que je vais sûrement oublier quelqu'un, mais je vous remercie tous. Je vais ici me restreindre, sans comprendre très bien les limites.

Je dois commencer par Pierre. Merci de savoir me suivre par moments et de me forcer à aller au-delà de mes peurs d'autrefois. C'est comme ça que nous avons fait trois déménagements transatlantiques ensemble et fait d'une longue liste d'appartements notre hogar. Merci aussi d'être toujours critique sur mes décisions professionelles, même si des fois je ne fais pas attention à tes opinions.

Une bonne partie de ce mémoire est dédiée aux résultats que j'ai obtenus avec Steve Hurder, une collaboration qui va bien au-delà du cadre strictement professionel. Merci Steve pour l'accueil à Chicago, pour les après-midi d'hiver devant la cheminée chez toi, mais surtout merci de t'être engagé si fortement dans un projet qui semblait innocent il y a 8 ans, et que nous n'avons pas encore fini.

Le reste de ce mémoire est dédié à des résultats obtenus en collaboration avec Pierre Dehornoy. Merci Pierre de m'aprendre de belles mathématiques et merci pour un projet de recherche qui ne fait que commencer. Je te remercie aussi beaucoup d'avoir changé la date de la conférence à Moscou de janvier à avril, le voyage Mexico-Moscou en janvier aurait été très di cile pour moi.

Même si à ce jour nous ne sommes pas encore arrivés à écrire quelque chose ensemble, je veux remercier Alberto Verjovsky, pour toutes les heures sous son arbre à discuter. Je tiens également à remercier tous mes collaborateurs, tant ceux avec qui j'ai réussi à publier quelque chose, comme ceux avec qui nous avons compris que nos idées ne marchaient pas, ou ceux avec qui j'ai toujours l'espoir de résoudre un petit problème.

Je tiens à remercier Nalini Anantharaman d'avoir accepté d'être garante de cette habilitation, même si je lui ai demandé au dernier moment. Un grand merci à Amie Wilkinson, Christian Bonatti et Daniel Peralta Salas pour avoir accepté de rapporter sur mon mémoire. Je remercie aussi Christophe Bavard, Frédéric Le Roux et Étienne Ghys de faire partie du jury de mon habilitation.

Je dois dire que j'ai toujours bénéficié de conditions de travail exceptionnelles, je veux remercier les équipes administratives et tous les collègues des laboratoires où j'ai été.

Ce mémoire a été écrit au Mexique, où je travaille actuellement. Je tiens à remercier Yann Bugeaud, Thomas Delzant, Delphine Karleskind, Jessica Maurer-Spoerk, Géraldine Schverer et Grégory Thureau pour leur aide pour réaliser les démarches bureaucratiques à distance et pour leurs explications des étapes à franchir pour arriver à la soutenance.

Une force incroyable a détruit une petite partie de ma ville il y a quelques semaines. 32 ans avant (comptés jour pour jour) cette force avait détruit une grande partie de la même ville. Cette 3 4 fois elle m'a laissé sans maison et j'ai mis les dernières semaines à recréer une routine que j'apprécie plus que jamais. Heureusement, ce mémoire était déjà écrit, mais il me rappelera toujours les 19 septembre 2017 et 1985.

Mexico, 24 octobre 2017.

L'hélicité est définie dans la Section 2.1, il s'agit d'un invariant de conjugaison C 1 pour les flots qui préservent un volume.

Dans l'article (P8) nous établissons une équivalence entre deux modèles discrets de propagation de gaz dans un réseaux, appelés en anglais Lorenz lattice gases. Ils ont des propriétés remarquables. Notre papier donne une recette pour passer d'un modèle à un autre sur des réseaux réguliers du plan.

Tous mes papiers peuvent être consultés sur ma page web : www.matem.unam.mx/rechtman/publications.html même structure que l'ensemble minimal du cas générique. Comment étudier un ensemble minimal ? Quels sont les aspects importants ? Une première approche est de se faire une image de cet ensemble minimal, de le visualiser. Finalement c'est un ensemble plongé dans R 3 et 3 est encore une petite dimension. J'ai essayé dans ce mémoire d'expliquer l'image que nous nous sommes faite de cet ensemble, la section 1.3 donne une façon de comprendre comment il est structuré. Tout n'y est pas dit, j'ai décidé de ne pas rentrer dans certaines complications et détails. Il s'agit donc d'une image idéalisée de l'ensemble, qui permet de comprendre les autres aspects que nous avons décidé d'étudier.

Une seconde approche naturelle est la théorie de la forme. Introduite par Borsuk [Bor68], elle permet d'étudier certains aspects des ensembles plongés dans un espace euclidien, en étudiant des 11

À paraître dans les mémoires de la conférence 31st Summer Conference on Topology and its Applications.

Dans ce texte je présente des résultats tirés des papiers (P5), (P7) et (P9). Le papier (P10) est une compilation de questions autour des flots de Kuperberg, il est cité dans ce texte.

Je vais donc utiliser quelques lignes pour vous raconter les résultats des articles (P6) et (P8), avant d'introduire mon mémoire. Dans (P6) nous avons trouvé deux preuves alternatives de l'énoncé suivant, qui a été originalement prouvé par C. H. Taubes [START_REF] Taubes | An observation concerning uniquely ergodic vector fields on 3-manifolds[END_REF] en utilisant des invariants de Seiberg-Witten. Nous disons qu'un champ de vecteurs est strictement ergodique si son flot admet une unique mesure invariante et si cette mesure est un volume.

TH ÉOR ÈME. Soit X un champ de vecteurs sur S 3 strictement ergodique. Alors l'hélicité de X est nulle.

Introduction

Dans ce mémoire je présente des résultats reliés à l'étude des flots en dimension 3. Le mémoire est divisé en deux chapitres. Le premier est dédié à mon travail pour comprendre l'ensemble minimal du piège de Kuperberg, ou dit d'une autre façon, l'ensemble minimal des seuls exemples connus de flots lisses sans points fixes et sans orbites périodiques sur S 3 . Dans le second, j'explique comment construire une quantité appelée tronc associée à un flot sur S 3 muni d'une mesure invariante, qui est préservée par conjugaison topologique. Ce résultat s'inscrit dans la démarche consistant à trouver des invariants pour les flots provenant d'invariants pour les noeuds.

Les résultats présentés dans le premier chapitre ont été obtenus en collaboration avec Steve Hurder, avec qui j'ai commencé à collaborer lors de mon postdoctorat à Chicago. Nous nous sommes donné pour tâche de comprendre l'ensemble minimal du piège de Kuperberg. La première observation importante pour ce faire est que la construction dépend de certains choix, je cite É. Ghys [START_REF] Ghys | Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg)[END_REF] :

Par ailleurs, on peut construire beaucoup de pièges de Kuperberg et il n'est pas clair qu'ils aient la même dynamique.

Les choix donnent toujours un piège dont le flot est C OE , sans orbites périodiques et avec un unique ensemble minimal, mais nous ne savons pas si l'ensemble minimal est le même pour tous les choix possibles. En e et, une question qui reste ouverte est de savoir s'il y a des choix pour lesquels l'ensemble minimal est de dimension 1. Dans notre travail, nous imposons des hypothèses supplémentaires à la construction du piège qui nous permettent de montrer que l'ensemble minimal est de dimension topologique 2 et d'en déduire d'autres propiétés dynamiques du flot. Nous appelons ces choix génériques, car il s'agit de demander que deux propiétés de la construction soient de nature quadratique. Nous appelons les flots obtenus des flots de Kuperberg génériques. Je présente ces hypothèses brièvement dans la remarque 1.2.1 et avec plus de détails dans la section 1.6.

Il faut mentionner que dans la catégorie des flots linéaires par morceaux, Greg et Krystyna Kuperberg parviennent à construire un piège de Kuperberg dont l'ensemble minimal est de dimension 1 [START_REF] Kuperberg | Generalized counterexamples to the Seifert conjecture[END_REF]. Aussi, si un piège de Kuperberg est tel que son ensemble minimal est de dimension 1, l'ensemble minimal est contenu dans un esemble invariant de dimension topologique 2 qui a la Introduction suites de voisinages de plus en plus petits. Nous avons été guidés par une question de K. Kuperberg : est ce que la forme de l'ensemble minimal est stable ? Comme expliqué dans la section 1.4.1, nous savons que la réponse à cette question est négative (dans le cas d'un flot de Kuperberg générique). Le concept d'ensemble «movable» est un autre concept important dans la théorie de la forme. Nous ignorons si l'ensemble minimal des flots de Kuperberg génériques est «movable». Je conjecture que oui.

Nous comprenons donc comment visualiser cet ensemble minimal exceptionnel (c'est-à-dire transversement un ensemble de Cantor) de dimension topologique 2. Il est, par minimalité, localement homogène, mais il n'est pas globalement homogène. Comme expliqué dans le théorème 1.3.5, l'ensemble ne forme pas une lamination, mais il contient un ouvert dense de dimension 2 qui est une lamination L avec des feuilles ouvertes. Nous pouvons donc considérer la dynamique de cette lamination et la comparer à la dynamique du flot restreint à l'ensemble minimal.

Cet approche nous a permis de construire des pseudogroupes agissant, soit dans un rectangle presque transverse au flot, soit dans une transversale à la lamination L. Il s'agit, par leur nature et par la construction, de pseudogroupes di érents mais semblables, avec des ensembles de générateurs similaires. Le premier de ces pseudogroupes est décrit brièvement dans la section 1.4. En utilisant la notion d'entropie pour les pseudogroupes, introduite par Ghys, Langevin et Walczak [START_REF] Ghys | Entropie géométrique des feuilletages[END_REF], nous avons étudié ces pseudogroupes. Dans la section 1.4.2, j'explique quelques-uns des résultat obtenus.

Les pseudogroupes utilisés dans [P5] ont tous la propiété que le nombre de points séparés par des mots de longueur n (avec un ensemble de générateurs fixé) croît comme l'exponentielle de n -, pour un certain -oe (0, 1). Ils sont donc tous d'entropie nulle, mais d'entropie lente positive (les définitions sont données dans la section 1.4.2). Cette a rmation s'étend au flot restreint à l'ensemble minimal : le flot est d'entropie topologique nulle, mais a une entropie lente positive. Le fait que l'entropie topologique du flot soit nulle est aussi une conséquence d'un théorème de Katok [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for di eomorphisms[END_REF], comme remarqué par É. Ghys [START_REF] Ghys | Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg)[END_REF].

Avec l'objectif de trouver des flots à entropie topologique positive près des flots de Kuperberg, nous avons étudié des déformations de la construction du piège dans [P7]. Nous avons alors trouvé une famille C OE à un paramètre contenant un flot de Kuperberg et des flots à entropie positive. La construction et quelques idées des preuves forment le contenu de la section 1.5.

Le flot de Kuperberg est donc une bifurcation dans l'espace des flots sur une variété fermée de dimension 3. Il s'agit d'une situation non générique, mais ce flot doit probablement être entouré d'autres bifurcations parmi lesquelles il pourrait y avoir des bifurcations génériques. Je ne sais pas, pour le moment, si l'étude d'un voisinage du piège parmi les bifurcations est abordable.

Par ailleurs, les pièges à entropie topologique positive construites dans [P7] contiennent une famille dénombrable d'ensembles invariants dont la dynamique transverse est conjuguée à un fer à cheval. Comment ceux-ci dégénèrent-ils vers l'ensemble minimal du piège de Kuperberg ? Le temps de retour à ces ensembles tranverses invariants devient de plus en plus long quand on s'approche du piège de Kuperberg, mais j'ignore si c'est l'unique cause de la bifurcation. Il me semble donc intéressant de comprendre ce processus.

Les résultats du deuxième chapitre ont été obtenus en collaboration avec Pierre Dehornoy. Nous avons construit une quantité associée à un champ de vecteurs X muni d'une mesure invariante µ, qui est préservée par conjugaison. Nous appelons cette quantité un invariant de (X, µ). Ce problème est motivé par une observation de Helmholtz [Hel1858] : si X t est un champ de vecteurs non-autonome qui satisfait les équations d'Euler (dans le cas plus simple de ces équations) et si on note " t son flot, alors rot(X t ) est l'image sous " t de rot(X 0 ). Comme " t est un di éomorphisme qui préserve un volume, un invariant appliqué à rot(X 0 ) nous donne une quantité qui ne dépend pas du temps pour les solutions de l'équation d'Euler. L'invariant le plus connu est l'hélicité, il est défini quand la mesure invariante µ est un volume. Grâce aux travaux d'Arnold [START_REF] Arnold | The asymptotic Hopf invariant and its applications[END_REF], nous avons une interprétation topologique de cet invariant : dans S 3 (ou un domaine simplement connexe de R 3 ), l'hélicité coïncide avec le nombre d'enlacement asymptotique. Ce nombre est défini de la façon suivante. Prenons deux points x 1 , x 2 oe S 3 et deux nombres t 1 , t 2 oe R. On considère les courbes k(x i , t i ), pour i = 1, 2, formées par le segment d'orbite entre x i et " ti (x i ) suivi d'un arc (court) entre ces deux points. On peut montrer [Vog02] que pour presque toute paire de points et pour presque toute paire de nombres réels, on obtient deux courbes fermées simples et disjointes. Nous pouvons donc calculer leur nombre d'enlacement Nous avons décidé d'étudier un invariant des noeuds appelé le tronc, introduit par Ozawa [START_REF] Ozawa | Waist and trunk of knots[END_REF]. Cet invariant est construit en comptant le nombre de points d'intersection entre un noeud et les niveaux d'une fonction hauteur. Dans le cas de S 3 , une fonction hauteur a deux points singuliers et tout autre niveau est une sphère. Une adaptation naturelle au cas des champs de vecteurs est le flux géometrique à travers les niveaux de la fonction. Il s'agit de mesurer par rapport à une mesure invariante, le passage infinitésimal à travers la surface sans considérer l'orientation.

¸(k(x 1 , t 1 ), k(x 2 , t 2 )). L'hélicité est alors égale à ⁄ ⁄ 3 lim t1,t2aeOE ¸(k(x 1 , t 1 ), k(x 2 , t 2 )) t 1 t 2 4 dµdµ.
L'invariant qui en résulte est un invariant par conjugaison topologique, et il admet une interprétation asymptotique : dans le cas d'une mesure ergodique, il s'agit de la limite du tronc de k(x, t) divisé par t, pour presque tout x oe S 3 . Je présente dans le chapitre 2 les résultats que nous avons obtenus concernant cet invariant, en particulier, nous pouvons montrer qu'il n'est pas proportionnel à l'hélicité. Dans la section 2.1, j'ai décidé d'inclure le calcul qui nous permet de montrer cette dernière a rmation.

Introduction

Chapter 1

A minimal set

A 3-dimensional closed manifold has Euler characteristic zero, meaning that it admits a nonsingular vector field. There is one known way to build C OE or real analytic flows without fixed points and without periodic orbits that applies to any of these manifolds: using Kuperberg plugs. A plug allows to modify a flow inside a flow-box, trapping orbits and introducing at least a minimal set.

K. Kuperberg's construction appeared in 1993, published in 1994 [START_REF] Kuperberg | A smooth counterexample to the Seifert conjecture[END_REF]. The article was then followed by a Séminaire Bourbaki exposition by É. Ghys [START_REF] Ghys | Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg)[END_REF], a Sugaku lecture by S. Matsumoto [START_REF] Matsumoto | Kuperberg's C OE counterexample to the Seifert conjecture[END_REF] and a paper by Greg and Krystyna Kuperberg [START_REF] Kuperberg | Generalized counterexamples to the Seifert conjecture[END_REF]. Each of these papers gives di erent insights into the dynamics of Kuperberg flows, I will explain briefly some of them in Section 1.2.2. Regarding the minimal set of the flow inside the plug, it was known that there is only one minimal set and there where examples for which its dimension is 2.

In [P5], S. Hurder and I went into the details of these flows, trying to understand what the minimal set looks like. I will give here an informal description of it and cite the main results of our work, mainly related to its shape properties and some types of entropy of the flow. To my knowledge, our work gives the first explicit example of an exceptional minimal set of a flow that has topological dimension 2 and is not obtained from a suspension of a di eomorphism. This minimal set has amazing properties, some of them explained below. We developed a set of ad-hoc techniques for studying it. I ignore if any of these can be applied to other minimal sets.

The chapter is organized as follows. Sections 1.1 and 1.2 are a brief introduction to the problem and the results on Kuperberg flows previous to our work. Section 1.2 is divided into the original construction by K. Kuperberg explained in Section 1.2.1 and some known results on their dynamics other than aperiodicity explained in Section 1.2.2. In Section 1.3 and its subsections, I tried to give a picture of the minimal set. In Section 1.3.1, I start by explaining the structure of two special orbits of the flow, then in Section 1.3.2 I use these two orbits to decompose a dense subset of the minimal set whose dimension is 2. This is not a formal exposition since it avoids several minor complications and I refer for the proofs of the facts used to [P5].

In the paper [P5] we used several pseudogroups acting on a rectangle that is almost transverse to the flow of the Kuperberg plug to study the dynamics beyond aperiodicity. Even if in this text I don't present any proof using pseudogroups, I included in Section 1.4 the choice of the rectangle and some of the maps we studied in [P5]. These maps then appear in the discussions in Sections 1.4.2 and 1.5.

Section 1.5 corresponds to the results obtained in [P7]. Kuperberg flows have topological entropy zero, as a consequence of Katok's theorem on C 2 -flows [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for di eomorphisms[END_REF]. The question that motivated the results in [P7] was whether there are positive topological entropy flows arbitrarily near the Kuperberg flows. The answer is yes in the C 1 -topology by more general results on 3-dimensional 16 CHAPTER 1. A MINIMAL SET flows. Since the Kuperberg examples are explicit, it seemed that we could work in the C OE -topology. The answer is again yes: we found an explicit construction of a 1-parameter family containing a Kuperberg flow and flows of positive topological entropy. I give the construction highlighting the di erence with the original construction by K. Kuperberg and explain the main ideas in the proof.

The Seifert conjecture, a story beyond Wilson, Schweitzer and Kuperberg

In 1950, Seifert proved that vector fields close to a vector field tangent to the Hopf fibration have periodic orbits [Sei1950]. He asked whether any non-singular vector field on the three sphere S 3 had a periodic orbit, the positive answer to this question became known as the Seifert conjecture.

When studying a vector field on a manifold, the first relevant question is to know if it has zeros. To each isolated zero of the vector field we can associate an integer index, and the Poincaré-Hopf theorem tell us, that the sum of the indeces is equal to the Euler characteristic of the manifold. After Seifert result, there was the hope to find an analog of the Poincaré-Hopf theorem for periodic orbits. The result by F. W. Wilson mentioned below killed the hope to find such an index in dimension greater than 3, and then the results by P. Schweitzer and K. Kuperberg imply that there is no analog in dimension 3.

In 1966, F. W. Wilson built a plug that allows to obtain on any closed 3-manifold a non-singular vector field with a finite number of periodic orbits [START_REF] Wilson | On the minimal sets of non-singular vector fields[END_REF][START_REF] Percell | Plugging flows[END_REF] (the construction in the second paper is simpler). Moreover, this can be donne in any homotopy class of vector fields. That is, any homotopy class of non-singular vector fields contains a vector field with a finite number of periodic orbits. For manifolds of dimension strictly greater than 3, Wilson's construction provides a vector field without periodic orbits; at the place of the finite number of periodic orbits in dimension 3, in dimension n > 3 the construction gives a finite number of invariant n ≠ 2 dimensional tori endowed with an aperiodic flow. Thus on any closed manifold of dimension n > 3 with Euler characteristic zero, each homotopy class of non-singular vector fields contains a vector field without periodic orbits.

Back to dimension 3, after Wilson's result the problem was how to destroy those periodic orbits and there was a possible path: to build a plug without periodic orbits.

To fix ideas, let me define a plug. A 3-dimensional plug is a manifold P endowed with a vector field X satisfying the following properties: the manifold P is of the form D ◊ [≠2, 2], where D is a compact 2-manifold with boundary ˆD. Let ˆz be the vertical vector field on P , where z is the coordinate on [≠2, 2]. The vector field X on P must satisfy the conditions:

(P1) vertical near the boundary: X = ˆz in a neighborhood of ˆP ; thus, D ◊ {≠2} and D ◊ {2} are the entry and exit regions of P for the flow of X , respectively; (P2) entry-exit condition: if a point (x, ≠2) is in the same trajectory as (y, 2), then x = y. That is, an orbit that traverses P , exits just in front of its entry point; (P3) trapped orbit: there is at least one entry point whose entire forward orbit is contained in P ; we will say that its orbit is trapped by P and we call the set of entry points with trapped orbit the trapped set; (P4) tameness: there is an embedding i : P ae R 3 that preserves the vertical direction on the boundary ˆP . A plug is aperiodic if there is no closed orbit for X . After Wilson's result an aperiodic plug will allow to build a vector field without periodic orbits. Indeed, we can use such a plug to destroy the periodic orbits one by one. Consider one periodic orbit, it su ces to embed the aperiodic plug in a flow-box intersecting the periodic orbit in such a way that the periodic orbit gets trapped inside the plug, thus it will no longer be periodic. This can be done by conditions (P1), (P3) and 1.1. THE SEIFERT CONJECTURE (P4). Condition (P2) guarantees that there are no new periodic orbits after surgery: there are no periodic orbits inside the plug and if an orbit intersects the plug and is not trapped, the entry-exit condition implies that it will stay periodic or non-periodic after surgery. Repeating this process at most finitely many times, gives an aperiodic flow on any closed 3-manifold. The first aperiodic plug was built by P. Schweitzer [START_REF] Schweitzer | Counterexamples to the Seifert conjecture and opening closed leaves of foliations[END_REF] in 1974, though the flow is only C 1 . The aperiodic plug that is C OE (and even real analytic) was built by K. Kuperberg in 1993 [START_REF] Kuperberg | A smooth counterexample to the Seifert conjecture[END_REF]. In both cases, the use of the plugs gives a vector field that is homotopic (through non-singular vector fields) to the original one. Hence, the surgery does not changes the homotopy class of the vector field. Notice that almost 20 years passed between the two constructions, in the meantime J. Harrison managed to make a C 2 version of Schweitzer's plug [Har88] and there are a couple of papers trying to prove that it is impossible to build an aperiodic plug with a smooth flow. But how to prove that there are no aperiodic plugs in the C r category for r > 2?

Let me highlight one of the main di erences between Schweitzer's and Kuperberg's construction. Condition (P3) in the definition of a plug implies that a trapped orbit has to accumulate on an invariant set, thus a plug contains a minimal set for the flow. Schweitzer's construction starts with a minimal set: the main idea is to substitute the periodic orbits in the Wilson plug with two copies of the Denjoy minimal set. The di erentiability problem comes from this set. J. Harrison changed the embedding of the minimal set to make the flow C 2 . Kuperberg's construction focuses on destroying the periodic orbits of the Wilson plug, there is some minimal set in the plug, but not too much was known about it before our work. I will cite S. Matsumoto [START_REF] Matsumoto | Kuperberg's C OE counterexample to the Seifert conjecture[END_REF] to describe K. Kuperberg's construction:

We therefore must demolish the two closed orbits in the Wilson Plug beforehand. But producing a new plug will take us back to the starting line. The idea of Kuperberg is to let closed orbits demolish themselves. We set up a trap within enemy lines and watch them settle their dispute while we take no active part. After Schweitzer's construction the idea to prove that it was impossible to build a C OE aperiodic plug concentrates on the minimal set inside the plug (the starting point of his construction). I want to mention a paper by M. Handel [START_REF] Handel | One-dimensional minimal sets and the Seifert conjecture[END_REF], in which he proves that if the trapped orbits of a plug accumulate on a minimal set whose dimension is 1 and this is the only invariant set for the flow in the plug, then the minimal set is surface-like: the flow restricted to the minimal set is topologically conjugated to the minimal set of a flow on a surface. He considers also the case of a minimal set whose topological dimension is two, but he makes the assumption that there is a disk-like section of the minimal set. In Kuperberg's plug the minimal set is not the only invariant set of the plug, as proved by S. Matsumoto (see Theorem 1.2.3), and when it has topological dimension 2 it does not admits a disk-like section.

There is also a paper by R. J. Knill that embeds Denjoy minimal sets in C OE -flows on S 3 , but these are not isolated as any neighborhood contains periodic orbits [START_REF] Knill | A C OE flow on S 3 with a Denjoy minimal set[END_REF]. Hence, until CHAPTER 1. A MINIMAL SET K. Kuperberg's construction, the idea was to look at the possible minimal sets for an aperiodic plug.

It seems a good moment to comment on two (di cult) questions. The first one is on the dimension of the minimal set of the Kuperberg plug. We proved in [P5] that, under some extra assumptions on the construction, this set has topological dimension two. We do not know if there are smooth Kuperberg flows whose minimal set has dimension 1. I have tried to build them without success. Can one prove that a minimal set with dimension 1 and unstable shape cannot be isolated (meaning that any neighborhood should contain periodic orbits)? The answer to this question is yes if the minimal set is a solenoid as proved by E. S. Thomas [START_REF] Thomas | One-dimensional minimal sets[END_REF].

Secondly, consider the case of volume preserving flows on 3-manifolds. We are at the stage of knowing that there are examples with a finite number of periodic orbits and C 1 examples without periodic orbits on any closed 3-manifold. These were built by G. Kuperberg [START_REF] Kuperberg | A volume-preserving counterexample to the Seifert conjecture[END_REF]. Is it possible to build a C OE volume preserving aperiodic plug? Can we a priori say something about its minimal or invariant sets?

Kuperberg's construction and previous results

As mentioned above, K. Kuperberg's idea is to destroy the periodic orbits in the modified Wilson's plug using the plug itself, or let the periodic orbits demolish themselves. So I start explaining how to build Wilson plug (actually a modified version of the original plug). The construction of the self-insertions that destroy the periodic orbits is explained in Section 1.2.1.

Consider the rectangle R = [1, 3] ◊ [≠2, 2] = {(r, z) | 1 AE r AE 3 & ≠ 2 AE z AE 2}.
Choose a C OE -function g : R ae [0, g 0 ] for g 0 > 0, which satisfies the "vertical" symmetry condition g(r, z) = g(r, ≠z). Also, require that g(2, ≠1) = g(2, 1) = 0 and that g(r, z) > 0 otherwise. Next, choose a C OE -function f : R ae [≠1, 1] which satisfies the following conditions:

(W1) f (r, ≠z) = ≠f (r, z) [anti-symmetry in z]. (W2) f (›) = 0 for › near the boundary of R. (W3) f (r, z) Ø 0 for ≠2 AE z AE 0. (W4) f (r, z) AE 0 for 0 AE z AE 2. (W5) f (2, ≠1) = 1 and f (2, 1) = ≠1.
Next, define the manifold with boundary

W = [1, 3] ◊ S 1 ◊ [≠2, 2] ≥ = R ◊ S 1 (1.1)
with cylindrical coordinates x = (r, ◊, z). That is, W is a solid cylinder with an open core removed, obtained by rotating the rectangle R, considered as embedded in R 3 , around the z-axis. Extend the functions f and g above to W by setting f (r, ◊, z) = f (r, z) and g(r, ◊, z) = g(r, z), so that they are invariant under rotations around the z-axis. The modified Wilson vector field W on W is defined by

W = g(r, ◊, z) ˆz + f (r, ◊, z) ˆ◊ .
(1.2) Let t denote the flow of W on W. Observe that the vector field W is vertical near the boundary of W and is horizontal at the points (r, ◊, z) = (2, ◊, ±1). Also, W is tangent to the cylinders {r = cst.}. The flow t on the cylinders {r = cst.} is illustrated by Figure 1.3.

O i = {(2, ◊, (≠1) i )} [Periodic Orbits, i=1,2 ] ˆ≠ h W = {(r, ◊, ≠2)} [The Entry Region] ˆ+ h W = {(r, ◊, 2)} [The Exit Region]
Then O 1 is the lower boundary circle of the Reeb cylinder R and O 2 is the upper boundary circle. The flow t has exactly two periodic orbits O 1 and O 2 , the maximal invariant set inside the plug is the Reeb cylinder R and satisfies the entry-exit condition (P2) as a consequence of the symmetry in the function g and condition (W1). Finally, observe that the trapped set are the points with r = 2 in the entry region of W. The fundamental idea of the Kuperberg Plug is to construct two insertions of W into itself, in such a way that the two periodic orbits will be trapped by these self-insertions. Moreover, the insertions are made so that the resulting space K is again embedded in R 3 . A key subtlety of the construction arises in the precise requirements on these self-insertions. As with the construction of the modified Wilson Plug, the description of this construction in the works [Kup94, Ghys95, Mat95] is qualitative, as this su ces to prove the aperiodicity of the resulting flow. As explained in [P5], other properties of the dynamics of the flow t in the resulting plug K are strongly influenced by the precise nature of these maps, so some hypotheses were added to the construction. We call a plug satisfying these hypotheses a generic Kuperberg plug and its flow a generic Kuperberg flow. In Remark 1.2.1 at the end of this section I briefly explain the generic hypotheses and in Section 1.6 I give a compilation of the generic hypotheses.

After the embedding presented in

L ± 1 = L 1 ◊ {±2} , L ± 2 = L 2 ◊ {±2} . (1.3)
The next step is to define insertion maps ‡ i : D i ae W, for i = 1, 2, in such a way that the periodic orbits O i flow t intersect ‡ i (L ≠ i ) in points corresponding to W-trapped entry points for the Wilson plug W. Consider two disjoint arcs -Õ i in the inner boundary circle {r = 1}, that are in front of the arcsi when the plug is embedded as in Figure 1.5. For i = 1, 2, choose orientation preserving di eomorphisms ‡ i : -Õ i ae -Õ i and extend these maps to smooth embeddings ‡ i : D i ae W, as illustrated in Figure 1.7, which satisfy the conditions:

(K1) ‡ i (-Õ i ◊ z) = -Õ i ◊ z for z oe [≠2, 2], the interior arc -Õ i is mapped to a boundary arc -Õ i ; (K2) for D i = ‡ i (D i ), D 1 fl D 2 = ÿ; (K3) ‡ 1 (L ≠ 1 ) µ {z < 0} and ‡ 2 (L + 2 ) µ {z > 0}; (K4) For every x oe L i , the image ‡ i (x ◊ [≠2, 2]) is an arc contained in a trajectory of W; (K5) Each slice ‡ i (L i ◊ {z}
) is transverse to the vector field W, for all ≠2 AE z AE 2;

KUPERBERG'S CONSTRUCTION AND PREVIOUS RESULTS

(K6) D

i intersects the periodic orbit O i and not O j , for i " = j. For i = 1, 2, the components of the boundary of the embedded regions The embeddings ‡ i are also required to satisfy two further conditions, which are the key to showing that the resulting Kuperberg flow t is aperiodic:

D i = ‡ i (D i ) µ W that are transverse to W are labeled by L ± i = ‡ i (L ± i ) . (1.4) Note that the arcs ‡ i (x ◊ [≠2, 2]) in condition (K3) are line segments from ‡ i (x ◊ {≠2}) oe L ≠ i to ‡ i (x ◊ {2}) oe L + i which
(K7) For i = 1, 2, the disk L i con- tains a point (2, ◊ i ) such that the image under ‡ i of the vertical seg- ment (2, ◊ i ) ◊ [≠2, 2] µ D i µ W is an arc of the periodic orbit O i of W. (K8) Radius Inequality: For all x = (r Õ , ◊ Õ , z Õ ) oe L i ◊ [≠2, 2], let (r, ◊, z) = ‡ i (r Õ , ◊ Õ , z Õ ) oe D i , then r Õ > r unless x = (2, ◊ i , z
). The Radius Inequality (K8), illustrated in Figure 1.8, is one of the most fundamental concepts of Kuperberg's construction. This is an "idealized" case, as it implicitly assumes that the relation between the values of r and r Õ is "quadratic" in a neighborhood of the special points (2, ◊ i ), which is not required in order that (K8) be satisfied. This "quadratic condition" is part of the generic hypotheses on the construction (see Remark 1.2.1 and Hypothesis 1.6.3).

The Radius Inequality (K8) is a monotone condition that allows to keep track of the behavior of the orbits in the Kuperberg plug. It is this condition that is violated in the construction in Section 1.5.

The embeddings ‡

i : L i ◊ [≠2, 2] ae W, for i = 1, 2
, can be constructed by first choosing smooth embeddings of the faces ‡ i : L ≠ i ae W so that the image surfaces are transverse to the vector field W on W and satisfy the conditions (K1), (K5) for z = ≠2, (K7) and (K8). Then we extend the embeddings of the faces L ≠ i to the sets L i ◊ [≠2, 2] by flowing the images using a reparametrization of the flow of W, so that we obtain embeddings of L i ◊ [≠2, 2] satisfying conditions (K1) to (K8), as pictured in Figure 1.7 for the bottom insertion.

CHAPTER 1. A MINIMAL SET

Finally, define K to be the quotient manifold obtained from W by identifying the sets D i with D i . That is, for each point x oe D i identify x with ‡ i (x) oe W, for i = 1, 2, as illustrated in Figure 1.9. The restricted t -flow on the inserted disk

D i = ‡ i (D i )
is not compatible with the image of the restricted t -flow on D i . Thus it is necessary to modify W on each insertion D i , by replacing the vector field W on the interior of each region D i with the image vector field, so that the dynamics in the interior of each insertion region D i reverts back to the Wilson dynamics on D i . As for the insertion of plugs, the flow has to be reparametrized near the boundary of the insertion so that the resulting flow is C OE . Let K be the resulting vector field and t its flow.

Figure 1.9: The Kuperberg Plug K REMARK 1.2.1 (The generic hypotheses). Under the name generic hypotheses, we added in [P5] a set of conditions to the constructions of the modified Wilson plug W and the Kuperberg plug K that allowed us to study the dynamics of the flows beyond aperiodicity. Some of these are technical assumptions and can be classified into two classes:

-The function g in the construction of the modified Wilson plug W is zero only at the points (2, ±1) of the rectangle and the speed at which it goes to zero near this points is specified in the generic hypotheses: we ask g to be a quadratic function of the distance to these points (in a small neighborhood). -The Radius Inequality (K8) is required to be quadratic near the special points, as suggested by Figure 1.8.

These assumptions are used to prove that the minimal set in the plug has topological dimension 2 (see Chapter 17 of [P5]), to give explicit computations of the topological entropy of the flow restricted to the minimal set (see Chapters 20 and 21 of [P5]) and to justify the description of the minimal set (see Chapter 18 of [P5]).

Recently, D. Ingebretson used the generic hypotheses to interpret the minimal set as the minimal set of an iterated function system and compute its Hausdor dimension [HI].

Notation and basic results on the dynamics

In this section, I introduce notations that will be used throughout this chapter. The notation used here is the same as in [P5]. Introduce the sets:

W Õ = W ≠ {D 1 fi D 2 } , " W = W ≠ {D 1 fi D 2 } .
(1.5)

The compact space " W µ W is the result of "drilling out" the interiors of D 1 and D 2 .

Figure 1.10: " W with some Wilson arcs For x, y oe K, we say that x ª K y if there exists t Ø 0 such that t (x) = y. Likewise, for x Õ , y Õ oe W, we say that x Õ ª W y Õ if there exists t Ø 0 such that t (x Õ ) = y Õ . Let • : W ae K denote the quotient map, which for i = 1, 2, identifies a point x oe D i with its image ‡ i (x) oe D i . Then the restriction • Õ : W Õ ae K is injective and onto. Let (• Õ ) ≠1 : K ae W Õ denote the inverse map, which followed by the inclusion W Õ µ W, yields the (discontinuous) map • ≠1 : K ae W. For x oe K, let x = (r, ◊, z) be defined as the W-coordinates of • ≠1 (x) µ W Õ . In this way, we obtain (discontinuous) coordinates (r, ◊, z) on K. In particular, let r : W Õ ae [1, 3] be the restriction of the radius coordinate on W, then the function is extended to the radius function of K, again denoted by r, where for

x oe K set r(x) = r(• ≠1 (x)).
The flow of the vector field W on W preserves the radius function on W, so x Õ ª W y Õ implies that r(x Õ ) = r(y Õ ). However, x ª K y need not imply that r(x) = r(y). The points of discontinuity for the function t ' ae r( t (x)) play a fundamental role in the study of the dynamics of Kuperberg flows.

Figure 1.10, copied from Ghys' paper [START_REF] Ghys | Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg)[END_REF], is fundamental for the understanding of the Kuperberg plug K. Indeed, an orbit in K can be chopped into segments of W-orbits, or in other words into segments of orbits in " W whose endpoints lie in the boundary ˆ" W. Thus, up to identification of L ± i with L ± i , Figure 1.10 contains all the information needed to follow the K-orbits. This is the technique used to study the dynamics inside K, as explained below or in any of the references [Ghys95, KK96, Kup94, Mat95, P5].

Let

ˆ≠ h K = • (ˆ≠ h W \ (L ≠ 1 fi L ≠ 2 )) and ˆ+ h K = • (ˆ+ h W \ (L + 1 fi L + 2
)) denote the bottom and top horizontal faces of K, respectively. That is, the entry and exit regions of K. Points x Õ oe ˆ≠ h W and y Õ oe ˆ+ h W are said to be facing, we write x Õ © y Õ , if x Õ = (r, ◊, ≠2) and y Õ = (r, ◊, 2) for some r and ◊. The entry/exit property of the Wilson flow is then equivalent to the property that

x Õ © y Õ if [x Õ , y Õ ] W is an orbit from ˆ≠ h W to ˆ+ h W whenever r(x Õ ) " = 2.
There is also a notion of facing points for x, y oe K, if either of two cases are satisfied:

-

For x = • (x Õ ) oe ˆ≠ h K and y = • (y Õ ) oe ˆ+ h K, if x Õ © y Õ then x © y. -For i = 1, 2, with x = ‡ i (x Õ ) and y = ‡ i (y Õ ), if x Õ © y Õ then x © y.
Consider the embedded disks L ± i µ W defined by (1.4), which appear as the faces of the insertions in " W in Figure 1.10 that are transverse to the vector field W. Their images in the quotient manifold K are denoted by:

E 1 = • (L ≠ 1 ) , S 1 = • (L + 1 ) , E 2 = • (L ≠ 2 ) , S 2 = • (L + 2 ) . (1.6) Note that • ≠1 (E i ) = L ≠ i , while • ≠1 (S i ) = L + i .
• ≠1 to a W-orbit in W flowing from ˆ≠ h W to ˆ+ h W.
The special points for the flow t are the images, for i = 1, 2,

p(i) = • (O i fl L ≠ i ) oe E i , p(i) = • (O i fl L + i ) oe S i .
(1.7)

Then p(i) © p(i) for i = 1, 2 and by the Radius Inequality (K8), we have r(p(i)) = r(p(i)) = 2 for i = 1, 2. The two K-orbits containing these points are the special orbits

S i for i = 1, 2. A W-arc is a closed segment [x, y] K µ K of the flow of K whose only transition points are the endpoints {x, y}. The open interval (x, y) K is then the image under • of a unique W-orbit segment in W Õ , denoted by (x Õ , y Õ ) W where • (x Õ ) = x and • (y Õ ) = y Let [x Õ , y Õ ] W denote the closure of (x Õ , y Õ ) W in " W, then we say that [x Õ , y Õ ] W is the lift of [x,
y] K and is an arc of W-orbit as in Figure 1.10. Note that the radius function r is constant along [x Õ , y Õ ] W .

The level function along a K-orbit indexes the discontinuities of the radius function. Given x oe K, set n

x (0) = 0 and for t > 0 define

n x (t) = # {(E 1 fi E 2 ) fl s (x) | 0 < s AE t} ≠ # {(S 1 fi S 2 ) fl s (x) | 0 < s AE t} . (1.8) That is, n x (t)
is the total number of secondary entry points, minus the total number of secondary exit points, traversed by the flow of x over the interval 0 < s AE t. Thereafter, n

x (t) changes value by ±1 at each t > 0 such that t (x) is a transition point and whether the value increases or decreases, indicates whether the transition point is an entry or exit point.

The level function is the main tool for studying the dynamics of t . Consider an orbit in K and assume that it is not a Wilson orbit, meaning that it contains a certain number of transition points. Cutting the orbit at these points gives a sequence of W-arcs that lift to pieces of orbit in " W (as in Figure 1.10). We understand completely all the possible pieces in " W, thus what is important is to understand how these pieces concatenate to form an orbit in K.

The level function measures how "deep" the orbit goes: how many entries has passed, without crossing the corresponding exit points. This allows to follow the orbit and understand the behavior of the radius coordinate. In Section 1.3.2 the level function is used in a fundamental way to understand the structure of the minimal set.

Clearly the main result on K is: THEOREM 1.2.2. K endowed with the vector field K is an aperiodic plug.

The proof uses strongly the level function. The Radius Inequality (K8) is only used for proving that there are no periodic orbits inside the plug. We refer to any of the papers [Ghys95, KK96, Kup94, Mat95, P5] for a proof.

I also like to state a result by S. Matsumoto on the dynamics of the plug (see Proposition 7.5 of [P5] for a proof). THEOREM 1.2.3. The trapped set of K contains a set with non-empty interior.

In particular, Theorem 1.2.3 implies that there is a "big" invariant set. In the next section I describe the minimal set, first as the closure of the special orbits, then as the closure of a 2-dimensional set M 0 obtained by flowing the Reeb cylinder • (R) in K. The proof of Theorem 1.2.3 implies that there is an invariant set that is larger than M 0 . In Chapter 16 of K we describe the orbits of this invariant set.

The minimal set under the generic hypotheses

The aim of this section is to give a description of the minimal set under the generic hypotheses (see Remark 1.2.1 and Section 1.6). I start by explaining how the special orbits, that is the orbits obtained after the Kuperberg surgery from the periodic orbits of the Wilson plug, behave. As remarked previously, an orbit in the Kuperberg plug K is a concatenation of segments of Wilson orbits whose endpoints are in the boundary of " W (see Figure 1.10). These segments of W-orbits are quite simple, thus to understand orbits in K one has to understand the rules of concatenation. The two special orbits have similar structure, I just treat the case of one of them. Once the structure of these is orbits is settled, the structure of the minimal set comes naturally.

The objective is to give a visual explanation of the objects mentioned above. By doing so, I will skip some complications. Some of the minor ones are the appearance of "bubbles" in the minimal set (see Chapters 15 and 18 of [P5]), the fact that the construction of K is not completely symmetric (see Chapter 9 of [P5] for a proof that the non-symmetry has minor e ects on the dynamics). The more important one, is not proving that the generic hypotheses imply that the minimal set has topological dimension 2, as proved in Chapter 17 of [P5].

The special orbits

The minimal set of the plug K is obtained as the closure of any of the special orbits S 1 or S 2 , these are the orbits in K that contain the arcs

O Õ i = O i fl W Õ of the periodic orbits of the Wilson plug W. That is S i = K(O Õ i ) = { t (x) | x oe O Õ i , t oe R}, observe that the endpoints of O Õ
i are the special points p(i) and p(i) defined in (1.7), for i = 1 and 2. What follows explains why the periodic orbits of Wilson are not longer periodic after the surgery, but it is not a proof of the aperiodicity of the plug. The explanation implies also that S 2 µ S 1 and when carried out for S 2 (I do not give the details) we get that S 1 = S 2 is a minimal set. Proving that this set is the only minimal set in the plug is a harder task: it involves understanding the asymptotic behavior of every orbit that is entirely contained in K. In Proposition 7.1 of [P5], it is proved that the Ê-limit of every point in K whose orbit stays in K contains S 1 , while its --limit contains S 2 . Thus = S 1 = S 2 is the unique minimal set.

We recall first from Figure 1 i for i = 1, 2. Since this cylinder intersects both insertions, in Figure 1.11 I erased two regions of the rectangle, one intersecting O 1 and the other O 2 , corresponding to each insertion. Observe that these two regions are basically rectangles, with two of their sides tangent to the vector field and two transverse to the vector field. The transverse sides are either in the entrance of the insertions L ≠ i or in the exit of the insertions L + i , for i = 1, 2. The orbit segment O Õ 1 intersects the bottom entrance L ≠ 1 and the bottom exit L + 1 . Condition (K7) implies that the intersection with L ≠ 1 is at the special point p(1) that is identified with the point

p Õ (1) = (2, ◊ 1 , ≠2) oe L ≠ 1 fl{r = 2}. That is ‡ 1 (p Õ (1)) = p(1), • (p(1)) = • (p Õ (1)
) and r(p Õ (1)) = 2. By abuse of notation, let p(1) be the corresponding point in E 1 µ K. Likewise, the intersection of O Õ 1 with L + 1 is at the special point p(1) that is identified with the point p

Õ (1) = (2, ◊ 1 , 2) oe L + 1 fl {r = 2}. That is ‡ 1 (p Õ (1)) = p(1), • (p(1)) = • (p Õ (1)
) and r(p Õ (1)) = 2. By abuse of notation, let p(1) be the corresponding point in S 1 µ K.

In general, I will use the same notation for points in L ± i and their images under • in E i and S i for i = 1, 2. The corresponding points in L ± i are marked with a prime. The positive K-orbit of a point in • (O Õ 1 ) continues after crossing E 1 at p(1) as a trapped W-orbit in the cylinder {r = 2}, that is the image under • of the W-orbit of p Õ (1). Since the W-orbit of p Õ (1) accumulates on O 1 , it intersects infinitely many times L ≠ 1 . Then the image under • of this orbit intersects E 1 = • (L ≠ 1 ) infinitely many times. Let p(1; 1, 1) be the first intersection. Denote by p Õ (1; 1, 1) oe L ≠ 1 the point in the entrance of W such that • (p Õ (1; 1, 1)) = p(1; 1, 1), then the Radius Inequality (K8) implies that r(p Õ (1; 1, 1)) > 2.

We need the following important result (we refer to Propositions 6.5 and 6.7 of [P5]): PROPOSITION 1.3.1. Let x be a primary or secondary entry point with r(x) > 2, let x be the exit point facing x (x © x). Then x is in the same K-orbit as x and the K-orbit of x contains the ordered collection of W-arcs in the W-orbit of x Õ that are in W Õ , where

• (x Õ ) = x.
The proposition implies that the K-orbit S 1 will make a (a priori) complicated trajectory from p(1; 1, 1) oe E 1 µ K before reaching S 1 at the facing point p(1; 1, 1) (this point is not necessarily the first intersection of the orbit with the secondary exit region S 1 ). Condition (K4) implies that p(1; 1, 1) and p(1; 1, 1) belong to the same Wilson orbit that is contained in the cylinder {r = 2}.

The above description can be applied repeatedly to S 1 , always in the positive direction. From p(1; 1, 1) the orbit makes a turn around the cylinder {r = 2} and intersects E 1 at a point p(1; 1, 2) that is above p(1; 1, 1) and below p(1), as illustrated in Figure 1.23. Since r(p(1; 1, 2)) > 2, Proposition 1.3.1 applies once more implying that p(1; 1, 2) oe S 1 facing p(1; 1, 2) belongs to the orbit S 1 . We can then continue forever: from p(1; 1, 2) follows a W-arc that makes a turn around the cylinder {r = 2} to a point p(1; 1, 3) oe E 1 , with r(p(1; 1, 3)) > 2. Then Proposition 1.3.1 guarantees that p(1; 1, 3) oe S 1 is in S 1 , etc.

Since the W-orbit of p Õ (1) accumulates on the periodic orbit O 1 , the forward

K-orbit of p(1) oe E 1 µ K accumulates on • (O Õ 1 ) µ S 1 .
That is, recursively we obtain points p(1; 1, ¸) for ¸Ø 1 in E 1 that belong to the special orbit S 1 and accumulate on p(1). The Radius Inequality (K8) implies that r(p(1; 1, ¸)) is an decreasing sequence converging to 2 as ¸ae OE (at least for ¸big enough the sequence is decreasing). REMARK 1.3.2. We have obtained that the Ê-limit of the points in S 1 , denoted by Ê(S 1 ), contains O Õ 1 and thus contains S 1 . In other words, S 1 µ Ê(S 1 ).

Figure 1.12: The intersection of the orbit of p Õ (1; 1, 1) with the cylinder {r = r 1 } Now let us fill the gap between p(1; 1, 1) and p(1; 1, 1), the same explanation applies to the segments of orbit between p(1; 1, ¸) and p(1; 1, ¸) but gets increasingly complicated as ¸grows since the radius of the starting point p(1; 1, ¸) is closer to 2. I will give an explanation for ȩqual to 1 and 2. As entry point p(1; 1, 1) has radius greater than 2, thus the point p Õ (1; 1, 1) oe L ≠ 1 µ W has radius r 1 greater than 2. We have two possibilities, either the cylinder {r = r 1 } intersects the insertions or it does not. Assume that it does not as in Figure 1.12. Then the entire Wilson orbit of p Õ (1; 1, 1) is contained in W Õ and hence the image under • of the W-orbit of p Õ (1; 1, 1) is a segment of K-orbit joining p(1; 1, 1) oe E 1 to p(1; 1, 1) oe S 1 . From p(1; 1, 1) the orbit S 1 makes one turn around the cylinder {r = 2} and it intersects E 1 at the point p(1; 1, 2). Proposition 1.3.1 implies that p(1; 1, 2) oe S 1 belongs to the orbit S 1 . I will now explain how to fill the gap between these two points. The Radius Inequality (K8) implies that r 2 = r(p(1; 1, 2)) > 2. We have the same two possibilities as above, either the cylinder {r = r 2 } intersects the insertions or it does not (here I will assume that the insertions are symmetric in the sense that either the cylinder intersects both insertions or none). The latter case is the simple one as treated for p(1; 1, 1). Now assume that the cylinder {r = r 2 } intersects both insertions and also that the W-orbit of p Õ (1; 1, 2) intersects both insertions, as in Figure 1.13. Clearly the image under • of the arcs above does not completely fills the gap between p(1; 1, 2) and p(1; 1, 2), since it is not a connected set. To do so, we need to understand what happens CHAPTER 1. A MINIMAL SET between the endpoint of an arc A j and the starting point of the following arc A j+1 .

Figure 1.14: The intersection of the orbit of p Õ (1; 1, 2; 1, 1) with the cylinder {r = r 2,1 }

Let us follow the orbit from p(1; 1, 2) = • (p Õ (1; 1, 2)). We start by the image under • of the first arc A 1 . The assumptions imply that the endpoint of this arc is a point in E 1 that we denote by p(1; 1, 2; 1, 1). Let p Õ (1; 1, 2; 1, 1) oe L ≠ 1 be the corresponding point in the entrance of W. Observe that the Radius Inequality (K8) implies that r 2 < r 2,1 = r(p(1; 1, 2; 1, 1)). Again there are two possibilities: either the cylinder {r = r 2,1 } intersects the insertions or not. If not, the image under • of the W-orbit p Õ (1; 1, 2; 1, 1) goes from p(1; 1, 2; 1, 1) to the facing point p(1; 1, 2; 1, 1) oe S 1 joining A 1 to A 2 . If yes we have to iterate the description. Since each time that the orbit intersects a secondary entry point the radius grows, eventually it gets to a cylinder that does not intersects any of the insertions. So assume, to make the discussion shorter, that we are in the second case, as in Figure 1.14.

Recapitulating, under the assumptions made, we have p(1; 1, 1)

• (W≠orbit) ≠≠≠≠≠≠≠ae p(1; 1, 1) W≠arc ≠≠≠≠ae p(1; 1, 2) • (A1)
≠ ≠≠≠ ae p(1; 1, 2; 1, 1)

• (W≠orbit)
≠≠≠≠≠≠≠ae p(1; 1, 2; 1, 1), where p(1; 1, 2; 1, 1) oe S 1 is the starting point of the arc • (A 2 ) and it is facing p(1; 1, 2; 1, 1) by condition (K4). Now • (A 2 ) has its starting point in S 1 and its endpoint either in E 1 or in E 2 (in Figure 1.12 the endpoint of A 2 is in L ≠ 1 with • (L ≠ 1 ) = E 1 ). Repeating the above description we fill in the orbit segment between the endpoint of • (A 2 ) and the starting point of • (A 3 ). We can thus join the collection of arcs • (A j ) and find the K-orbit segment from p(1; 1, 2) to p(1; 1, 2). This finishes the description of the forward K-orbit of points in O Õ 1 .

The above description applies without any significant changes to the backward orbit of O Õ 1 . Indeed the backward orbit of a point in O Õ 1 intersects L + 1 in the special point p(1) facing p(1) as in Figure 1.11. This point is identified with p Õ (1) oe L + 1 that has radius 2 and the intersection of the backward W-orbit of this point with W Õ is mapped under • to parts of S 1 . Since this W-orbit accumulates in backward time on O 2 , then

• (O Õ
2 ) is contained in the --limit of S 1 and thus S 2 µ -(S 1 ) µ S 1 . Also, recursively, in the backward direction, we can fill the gaps to describe the entire orbit. This is a first explanation of how the orbit S 1 is composed. In the rest of this section, always add "the image under • " to the sets considered. I will explain basically the same construction of S 1 , but in a new set of diagrams. For this we consider the level sets of S 1 . We need the following result (see Proposition 10.1 in [P5]).

PROPOSITION 1.3.3. For i = 1, 2, there is a well-defined level function

n 0 : S 1 fi S 2 ae N = {0, 1, 2, . . .},
(1.9)

such that n ≠1 0 (0) = • (O Õ 1 fi O Õ 2 ).
CHAPTER 1. A MINIMAL SET point in the entrance of the insertions E 1 fi E 2 , Proposition 1.3.1 implies that they will contain the facing points in the exit of the insertions S 1 fi S 2 . I represent in Figure 1.16 these orbits as the border of horizontal tongues, because of their shape inside W (as in Figures 1.2 and 1.3). Again the dotted parts are the segments that are not in the intersection with W Õ and thus do not belong to S 1 . As we travel downwards along the two level one W-orbits the tongues that appear at level 2 get longer, since the radius of their starting point gets closer to 2. This also implies that, eventually, the tongues intersect the insertions. In Figure 1.16 (and Figure 1.17 below), the assumption is that the W-orbit of p Õ (1; 1, 1) does not intersects the insertions and the W-orbit of p Õ (1; 1, 2) intersects both insertions. Then the intersection of the W-orbit of p Õ (1; 1, 2) with W Õ is composed of the arcs A j for 1 AE j AE k and k finite. In Figure 1.16 the value of k is 3. At any level n greater than 2, we consider the W-orbits of the points in the intersection of the Figure 1.17: The tree diagram for S 1 with marked tips

W-orbits at level n ≠ 1 with the entry regions E 1 and E 2 . Recursively we obtain the diagram, in Figure 1.17, representing levels 0 to 4. This situation repeats again at any level, adding to the complexity of the diagram in Figure 1.17.

Observe that this set of diagrams explain the concatenation of pieces of Wilson orbits that form the special orbit S 1 , but do not reflect how the orbit is embedded in K. The embedding is quite complicated by the nature of the maps ‡ i , for i = 1, 2. In particular, the width of the tongues that appear is roughly the same and equals the width of the Reeb cylinder, as a consequence of the shape of Wilson orbits. I will come back to this point in Section 1.3.2.

In Figure 1.17, I marked a set of points: every finite W-orbit intersects the annulus {z = 0} at a single point. Condition (W1) implies that at this point the vector field W is vertical and by condition (K3) this point is in W Õ (since all the finite Wilson orbits we are considering have radius greater than 2). For a finite W-orbit we will call this point its tip.

Consider thus the set of tips contained in S 1 and their lifts to W Õ . We obtain a countable set of points in the annulus {z = 0}, whose radii are arbitrarily close to 2. Hence the closed set S 1 contains points in • (R Õ ) the image of the Reeb cylinder in K. I finish this section pointing out some conclusions: -S i µ S j for all i, j = 1, 2, thus S 1 = S 2 = is the minimal set. -contains points in • (R Õ ). We are faced with a dichotomy: either the tips in S 1 fi S 2 accumulate on a Cantor set in the circle {z = 0, r = 2} contained in • (R Õ ) or they accumulate on the whole circle. Theorem 17.1 in [P5] states that under the generic hypotheses (see Remark 1.2.1 and Section 1.6), the tips accumulate on the whole circle, implying that has topological dimension 2. A description of this set, more precisely a dense dimension 2 subset of is given in the following section. We do not know if there is a smooth Kuperberg plug for which the minimal set has dimension 1. In the piecewise linear category, G. Kuperberg and K. Kuperberg [START_REF] Kuperberg | Generalized counterexamples to the Seifert conjecture[END_REF] construct a plug with a 1-dimensional minimal set.

M 0 , a dense subset of the minimal set

In this section I describe a dense dimension 2 subset of for a generic Kuperberg flow, that is denoted by M 0 . The set M 0 is obtained by flowing inside K the notched Reeb cylinder • (R Õ ) and the description is very similar to the decomposition of S 1 by levels explained in Figures 1.15, 1.16 and 1.17. Before starting, I need to introduce two types of surfaces in W: finite and infinite propellers.

For a moment, we will work in the Wilson plug. Consider a curve " in the entry region ˆ≠ h W of the plug. We parametrize this curve as "(t) with t oe [0, 1] and we assume that:

r("(0)) = 3; -r("(1)) < r("(t)) for t oe [0, 1); -r("(1)) Ø 2. The propeller P " is obtained by flowing " inside the Wilson plug. In the following figures we assume that " is transverse to the cylinders {r = cst.}, just to make the figures simpler. I describe next the surface P " .

Figure 1.18: Finite propeller inside W If r("(1)) > 2 describing P " is a simple game. Indeed the W-orbit of every point in " is finite, as it exits the plug in some finite time. The finite propeller P " is the compact surface composed by the finite orbits of the points "(t) for t oe [0, 1]. It has the shape of a tongue that turns in the positive ◊-direction as in Figure 1.18. Its boundary is composed by " in the entry region of W, the facing curve " in the exit region of W, the orbit of "(0) that is just a vertical segment in the vertical boundary of W and the orbit of "(1). Observe that the orbit of "(1) gets longer as the radius of "(1) tends to 2.

If r("(1)) = 2 it is not so simple since the forward orbit of "(1) is trapped. We consider in this case the forward orbits of the points in the curve " and the backward orbits of the points in the facing curve ", for all times for which the orbits are defined. Equivalently, we consider the union of finite propellers P "s for s oe [0, 1) and " s µ " the curve obtained as "([0, s]) and the two semi-infinite orbits: the forward orbit of "(1) and the backward orbit of the facing point "(1). The

CHAPTER 1. A MINIMAL SET infinite propeller P

" is an open surface that turns infinitely many times around the Reeb cylinder R. Observe that we do not consider its closure that consists of the union of P " and R.

such that n ≠1 0 (0) = • (R Õ ).
Let M k 0 µ M 0 be the set of points at level at must k, I describe next these sets for k = 0, 1, 2.

The level zero set is the image under • of the notched Reeb cylinder R Õ = R fl W Õ . In W Õ it is easy to visualize it (as in Figure 1.19), in K it is a little bit more di cult. Figure 1.20 is an attempt to represent this set as it is embedded in K µ R 3 . What the map ‡ 1 does to this cylinder is to choose a vertical segment in the cylinder ({◊ = ◊ 1 } in condition (K7)) and turns it to make it tangent to part of the boundary of the cylinder that got erased by the lower notch. Analogously, ‡ 2 chooses a vertical segment in the cylinder ({◊ = ◊ 2 } in condition (K7)) and turns it to make it tangent to part of the boundary of the cylinder that got erased by the upper notch. The directions of the flow force the two turns to be in opposite directions.

Observe that R Õ fl L ≠ 1 is a vertical line that we call the curve " (as in Figure 1.19) and let " Õ be the corresponding curve in L ≠ 1 (as in Figure 1.21). That is ‡ 1 (" Õ ) = " and • (" Õ ) = • ("). Analogously we have a curve ⁄ oe L ≠ 2 and The set M 2 0 consists of M 1 0 union the image under • of a countable set of finite propellers. Indeed each notch in P Õ " or P Õ ⁄ has to be filled with a propeller. The intersection of • (P Õ " ) with E i for i = 1, 2 consist of a countable family of curves, that by the Radius Inequality (K8) are contained in the set {r > 2}. Each one of these curves in E 1 corresponds to a point p(1; 1, ¸) for ¸Ø 1 and unbounded. The first two curves are illustrated in Figure 1.23. Each curve in

⁄ Õ oe L ≠ 2 . Since L ≠
E i is mapped under ‡ ≠1 i to a curve in L ≠ i , for i = 1, 2, that generates a finite propeller in W.
The image under • of these finite propellers is contained in M 2 0 . Analogously for • (P Õ ⁄ ). Each of these propellers adds a finite tongue to Figure 1.22, that gets longer as we travel along the level one infinite propellers • (P Õ " ) and • (P Õ ⁄ ). The finite propellers might have notches, since if they are long enough they will intersect the insertions. The tongues at level 1 are the horizontal tongues attached to the flattened infinite propellers in Figure 1.24 (where just • (P Õ " ) and some of its ramifications are illustrated, the upper side corresponding to • (P Õ ⁄ ) is analogous). The level three set M 3 0 is obtained from M 2 0 by filling the notches of the level 2 propellers with finite propellers. Analogously M k 0 is obtained from M k≠1 0 by filling the notches of the level k ≠ 1 propellers with finite propellers. This process continues and gives the surface M 0 . Under the generic hypotheses, the closure of M 0 is the minimal set . This means that the "boundary" of M 0 that consists of the orbits S 1 and S 2 is dense in M 0 . In Chapter 18 of [P5] we prove that = M 0 is a zippered lamination, roughly meaning that the interior has the structure of a lamination (transversely Cantor) and the boundary is dense in the interior of the leaves. The definitions of interior and boundary for are made precise in [P5]. I summarize the description in the last two sections with the following statement.

THEOREM 1.3.5 (Theorems 17.1 and 19.1 of [P5]). For a generic Kuperberg flow M 0 = is the minimal set of the plug. The set has topological dimension 2 and contains:

-an open 2-dimensional subset that forms a lamination with open leaves; -a 1-dimensional subset that is dense in the previous set. These two sets are disjoint.

Let me say a few words about the theorem. The lamination contains the interior of M 0 , obtained by flowing the open notched Reeb cylinder, thus M 0 minus the two special orbits. But M 0 is not transversely a Cantor set, while the lamination in Theorem 1.3.5 is. In Chapters 18 and 19 of [P5] we study the intersection of M 0 with a rectangle {◊ = cst.}, as the rectangle R 0 introduced in the The last two sets are not countable. The first set and the interior of the arcs in the third set are the intersection of the lamination in Theorem 1.3.5 with the rectangle, while the second set and the endpoints of the arcs in the third set are the intersection of the 1-dimensional subset in Theorem 1.3.5 with the rectangle.

REMARK 1.3.6. Observe from Figure 1.24 that M 0 has the structure of a tree whose branches correspond to the propellers and can be rooted by the choice of a point in • (R Õ ). The growth of this tree is then the growth of the leaves in the lamination of Theorem 1.3.5 and is studied in Chapter 14 of [P5]. The growth is closely related to the growth of words in the pseudoıgroup G ú K , introduced in the following section.

Pseudogroups

A tool introduced in [P5] that is used to study the dynamics of the flow, are the pseudogroups and pseudoıgroup acting on an almost transverse rectangle. In a pseudoıgroup we consider only compositions of a (symmetric) generating set, but we do not ask for the condition on union of maps to be in the pseudoıgroup, as for pseudogroups. This section corresponds to parts of Chapter 9 of [P5], where all details and proofs can be consulted. I introduce here the main maps, that arise in the discussions in Sections 1.4.2 and 1.5 related to entropy.

In the plugs W and K we can consider the rectangles {◊ = cst.}, one of them in K is depicted in Figure 1.25. We must be careful to chose a value ◊ 0 so that the rectangle R 0 = {◊ = ◊ 0 } µ W does not intersect neither D i nor D i for i = 1 and 2. Let R 0 be the corresponding rectangle in K, we use the same notation since • is bijective when restricted to such a rectangle. I will ask one more thing to this rectangle, that it lies between the two insertions as in Figure 1.25, so that from the upper part (z > 0) points flow to the upper insertion and from the bottom part (z < 0) points flow to the lower insertion.

Observe that, either in W or in K, the vector field is tangent to the rectangle near the boundary and also at the line {z = 0}. Let Ê i oe R 0 be the intersection of the periodic orbit O i with R 0 when considering the rectangle in W or the intersection of the arc O Õ i with R 0 when considering the rectangle in K, for i = 1, 2.

A finite W-orbit, that is an orbit going from the entrance to the exit of the plug W, if it intersects R 0 , it does so in a symmetric pattern with respect to the line {z = 0}. In this case, the construction of the flow implies that the intersection consists of a finite sequence of points contained in the vertical line of constant radius (the value of the radius is determined by the orbit). We have to consider two situations, either the orbit is tangent to R 0 at the line {z = 0} or not, in both situations the points in the intersection of the orbit with R 0 are paired: for each point (r, ≠z) in the intersection, (r, z) is also in the intersection.

We can define a first return map ' for the flow t , that will have discontinuities (I refer to Chapter 9 of [P5] for a complete discussion of the discontinuities and other properties of this map). The domain of ' is the set:

Dom( ' ) = {› oe R 0 | ÷ t > 0 such that t (›) oe R 0 and s (›) / oe R 0 for 0 < s < t} . (1.11)
The radius function is constant along the orbits of the Wilson flow, so that r( ' (›)) = r(›) for all › oe Dom( ' ). Also, note that the points Ê i for i = 1, 2 are fixed-points for ' and for all other points › oe R 0 with › " = Ê i , points climb up that is z( ' (›)) > z(›). Here I will consider just two types of first return maps for the flow t , instead of considering the general first return map that is too complicated for a succinct discussion. Observe first that if an arc of K-orbit [›, ÷] K with › and ÷ in R 0 does not contains transition points and its intersection with R 0 is only at its endpoints, then ÷ = ' (›). For i = 1, 2, let U " + i µ R 0 be the subset consisting of points › such that the K-arc [›, ÷] K contains a single transition point x oe E i and its intersection with R 0 is only at the endpoints › and ÷. Note that for such ›, we see from Figures 1.10 and 1.25, that its K-orbit exits the surface E i as the W-orbit of a point

x Õ oe L ≠ i with • (x Õ ) = x oe E i , flowing upwards from ˆ≠ h W until it intersects R 0 again. If the K-orbit of › enters E i but exits through S i before crossing R 0 , then › is not considered to be in the set U " + i
, since its orbits contains more than one transition point before returning to R 0 . Let " + i : U

" + i ae V " + i . As the K-arcs [›, ÷] K defining " + i do not intersect A, the restricted map " + i is continuous. The sets U " + i and V " + i
are sketched in the left-hand side illustration in Figure 1.26.

For i = 1, 2, let U

" ≠ i µ R 0 be the subset of R 0 consisting of points › such that the K-arc [›, ÷] K contains a single transition point x oe S
i and its intersection with R 0 is only at the endpoints › and ÷. Then let " ≠ i : U

" ≠ i ae V " ≠ i . Again, as the K-arcs [›, ÷] K defining the maps " ≠ i do not intersect A, the restricted map " ≠ i is continuous. The sets U " ≠ i and V " ≠
+ 1 , " + 2 , " ≠ 1 , " ≠ 2 } in Á U " + i
with R 0 is thus a region containing points with r-coordinate equal to 2 and since these points climb slower than other points, the region folds at {r = 2}. Similar considerations apply to the maps " ≠ i for i = 1, 2. Observe that each map " + i for i = 1, 2 corresponds to a flow through a transition point which increases the level function n

x (t) by +1, so the inverse of " + i decreases the level function by ≠1. The map " ≠ i decreases the level by ≠1 and its inverse increases the level by +1. Next, define the subset of R 0 contained in the domain of the Wilson map ' :

R ú 0 = {(r, ◊ 0 , z) oe Dom( ' ) | r Ø 2} \ {Ê 1 , Ê 2 } Proposition 1.3.1 implies (but it is not completely trivial) that there is a subset U  of R 0 that contains R ú 0 such that the restriction of ' to U  is a map representing part of the dynamics of the Kuperberg flow: if ' is the first return map to R 0 of the Kuperberg flow, then  = ' | U  is a power of ' (I refer to Lemma 9.6 in [P5]).
We have five special elements {" + 1 , " ≠ 1 , " + 2 , " ≠ 2 , Â} acting on subsets of R 0 , each of which reflects aspects of the dynamics of the flow t . DEFINITION 1.4.1. Let G ú K denote the collection of all maps formed by compositions of the maps {Id, " + 1 , " ≠ 1 , " + 2 , " ≠ 2 , Â} and their restrictions to open subsets in their domains.

Note that G ú K is not necessarily a pseudogroup since it might not be true that " : U ae V is a homeomorphism in G ú K whenever there is a cover of U by open sets {U a } aoeI such that "| Ua oe G ú K for all a oe I. The reason for not imposing this condition, is that many dynamical properties for flows admit corresponding versions for local maps defined by compositions of maps in this generating set, but not if we allow for arbitrary unions as for pseudogroups.

CHAPTER 1. A MINIMAL SET

The importance of G ú K is that its orbits are syndetic in the orbits of the flow. In [P5] this property is used to estimate the growth of the tree-like surface M 0 (Figure 1.24), give explicit computations for the entropy of the flow and the laminated entropy of the lamination in Theorem 1.3.5 (I refer to Chapters 14, 20, 21 and 22 of [P5]).

Shape of the minimal set

In this section, I present results in [P5] regarding topological properties of the minimal set for a generic Kuperberg flow. The space is compact and connected, so is a continuum. The natural framework for the study of topological properties of spaces such as is using shape theory. For example, Krystyna Kuperberg posed the question whether has stable shape? Stable shape is defined below and is about the nicest property one can expect for an exceptional minimal set. Theorem 1.4.5 below shows that does not have stable shape. The proof of this result follows from the equality = M 0 for a generic Kuperberg flow and the structure theory developed in Section 1.3.2. It is a constructive proof: we describe in Chapter 23 of [P5] a system of generators for the first homology groups of a shape approximation (an example of such a generator is illustrated in Figure 1.27).

Shape theory studies the topological properties of a topological space Z using a form of Čech homology theory. The definition of shape for a space Z embedded in the Hilbert cube was introduced by Borsuk [START_REF] Borsuk | Concerning homotopy properties of compacta[END_REF][START_REF] Borsuk | Theory of shape, Monografie Mat[END_REF]. We give a brief definition of the shape of a compactum Z embedded in a metric space X, following the work of Mardešić and Segal [START_REF] Mardešić | Shape theory: The inverse system approach[END_REF].

DEFINITION 1.4.2. A sequence U = {U ¸| ¸= 1, 2, . . .} is called a shape approximation of Z µ X if: 1. each U
¸is an open neighborhood of Z in X which is homotopy equivalent to a compact polyhedron;

U ¸+1 µ U

¸for ¸Ø 1 and their closures satisfy

‹ ¸Ø1 U ¸= Z.
Suppose that X, X Õ are connected manifolds, that U is a shape approximation for the compact subset Z µ X and U Õ is a shape approximation for the compact subset Z Õ µ X Õ . The compacta Z, Z Õ are said to have the same shape if the following conditions are satisfied:

1. There is an order-preserving map " : Z ae Z and for each n Ø 1 a continuous map f n : U "(n) ae U Õ n such that for any pair n AE m, the restriction

f n |U "(m) is homotopic to f m as maps from U "(m) to U Õ n . 2.
There is an order-preserving map  : Z ae Z and for each n Ø 1 a continuous map g n :

U Õ Â(n) ae U n such that for any pair n AE m, the restriction g n |U Õ Â(m) is homotopic to g m as maps from U Õ Â(m) to U n . 3. For each n Ø 1, there exists m Ø max{n, " ¶ Â(n)} such that the restriction of g n ¶ f Â(n) to U m is homotopic to the inclusion as maps from U m to U n . 4. For each n Ø 1, there exists m Ø max{n, Â ¶ "(n)} such that the restriction of f n ¶ g "(n) to U Õ m is homotopic to the inclusion as maps from U Õ m to U Õ n .
DEFINITION 1.4.3. Let Z µ X be a compact subset of a connected manifold X. Then the shape of Z is defined to be the equivalence class of a shape approximation of Z as above. Some examples of spaces with stable shape are compact manifolds and more generally finite CW -complexes. A less obvious example is the minimal set for a Denjoy flow on T 2 whose shape is equivalent to the wedge of two circles. In particular, the minimal sets of the aperiodic C 1 -flow constructed by Schweitzer have stable shape, [START_REF] Schweitzer | Counterexamples to the Seifert conjecture and opening closed leaves of foliations[END_REF]. The main result concerning generic Kuperberg flows is the following. THEOREM 1.4.5. The minimal set of a generic Kuperberg flow does not have stable shape.

Another important definition in shape theory is the following. DEFINITION 1.4.6. A compactum Z µ X is said to be movable in X if for every neighborhood U of Z, there exists a neighborhood U 0 µ U of Z such that, for every neighborhood W µ U 0 of Z, there is a continuous map Ï : U 0 ◊ [0, 1] ae U satisfying the condition Ï(x, 0) = x and Ï(x, 1) oe W for every point x oe U 0 .

The notion of a movable compactum was introduced by Borsuk [Bor69], as a generalization of spaces having the shape of an absolute neighborhood retract (ANR's) and is an invariant of the shape of Z. It is a standard result that a compactum Z with stable shape is movable. The movable property distinguishes between the shape of a Hawaiian earring and a Vietoris solenoid; the former is movable and the latter is not.

Showing the movable property for a space requires the construction of a homotopy retract Ï with the properties stated in the definition, whose existence can be di cult to achieve in practice.

CHAPTER 1. A MINIMAL SET

There is an alternate condition on homology groups, weaker than the movable condition, which is much easier to check. PROPOSITION 1.4.7. Let Z be a movable compacta with shape approximation U. Then the homology groups satisfy the Mittag-Le er Condition: For all ¸Ø 1, there exists p Ø ¸such that for any q Ø p, the maps on homology groups for m Ø 1 induced by the inclusion maps satisfy

Image {H m (U p ; Z) ae H m (U ¸; Z)} = Image {H m (U q ; Z) ae H m (U ¸; Z)} .
(1.12)

The above form of the Mittag-Le er condition can be used to show that the Vietoris solenoid formed from the inverse limit of coverings of the circle is not movable.

I can now state an additional shape property for the minimal set of a generic Kuperberg flow.

THEOREM 1.4.8. Let be the minimal set for a generic Kuperberg flow. Then the Mittag-Le er condition for homology groups is satisfied. That is, given a shape approximation U = {U ¸} for , then for any ¸Ø 1 there exists p > ¸such that for any q Ø p Image{H 1 (U

p ; Z) ae H 1 (U ¸; Z)} = Image{H 1 (U q ; Z) ae H 1 (U ¸; Z)}.
(1.13)

The strategy for the proof of Theorem 1.4.5 is to construct a shape approximation U for M 0 so that the conditions of Proposition 1.4.9 below are satisfied. This uses the properties of the level function n 0 defined in Proposition 1.3.4 and the double propellers introduced in Chapter 13 of [P5]. Double propellers provide a tool to build a set of neighborhoods of M 0 in which only propellers up to a certain fixed level can be distinguished. This provides a set of neighborhoods with su ciently nice behavior so that we are able to exhibit a generating set of the first homology groups of the shape approximation, one is sketched in Figure 1.27 (the dotted curves represent small jumps in E 1 fi E 2 fi S 1 fi S 2 ). The proof of Theorem 1.4.8 follows from the description of the homology groups.

PROPOSITION 1.4.9. Let U = {U ¸| ¸= 1, 2, . . .} be a shape approximation of Z µ X, such that for k > 0 -the rank of H 1 (U k ; Z) Ø 2 k , -for all ¸> k the rank of the image H 1 (U ¸; Z) ae H 1 (U k ; Z) is 3. Assume that for any shape approximation of V = {V ¸| ¸= 1, 2, . . .} the rank of the homology groups H 1 (V ¸; Z) is strictly greater than 3, then Z does not have stable shape.

To finish this section I will state two conjectures, the first one about the minimal set of the Kuperberg plug and the second one about general minimal sets. CONJECTURE 1.4.10. For a generic Kuperberg flow the minimal set is movable.

A proof needs a better description of the generators of the homology groups of a shape approximation. The second conjecture is motivated by the fact that the number of generators of the first homology groups of the shape approximation of M 0 proposed in [P5] grows exponentially with respect to the levels distinguished by the shape approximation and then also grows exponentially with respect to length. These generators are ' pseudo-orbits of the flow (that is pieces of orbits joined together by jumps of size bounded by '). CONJECTURE 1.4.11. If an exceptional minimal set of a flow is not shape equivalent to a solenoid and has unstable shape, then the rank of the first homology groups of a shape approximation grows exponentially.

In [START_REF] Barge | Pseudo-orbits and topological entropy[END_REF], the growth of the number of separated pseudo-orbits is related to the topological entropy of a flow. Thus if the conjecture is true and these generators can be interpreted as pseudo-orbits, a relation between shape and entropy type invariants could be explored. aspect that is not clear is what happens with all the periodic orbits in the plugs K ' for ' > 0. Is it just that their lengths blow up?

The bifurcation that is made explicit in Theorem 1.5.1 is highly not generic, but it might be surrounded by some type of generic bifurcations in the space of non-singular vector fields. Can we say something about nearby bifurcations?

Generic hypotheses

The construction of aperiodic Kuperberg flows involve multiple choices, which do not change whether the resulting flows are aperiodic, but do impact other dynamical properties of these flows. In this section, I present the assumptions made in the works [P5, P7], that are grouped under the word generic in the previous sections. In [P10] we discuss properties of the flow that depend on the generic hypotheses and discuss possible situations that can arise if these assumptions are dropped. This section thus gives a quick list of the hypotheses that were briefly explained in Remark 1.2.1. We discuss first the case for the traditional Kuperberg flows and afterward discuss the variations of the construction to obtain the family K ' described in Section 1.5.

Recall that the modified Wilson vector field on W is given in (1.2) by

W = g(r, ◊, z) ˆz + f (r, ◊, z) ˆ◊
where the function g(r, ◊, z) = g(r, z) is the function g : R ae [0, g 0 ] which is non-negative, vanishing only at the points (2, ±1) and symmetric about the line {z = 0}. The function f (r, ◊, z) = f (r, z) is assumed to satisfy the conditions (W1) to (W5). Figure 1.3 illustrates the dynamics of the flow of W restricted to the cylinders {r = cst.} in W, for various values of the radius. It is clear from these pictures that the interesting part of the dynamics of this flow occurs on the cylinders with radius near to 2 and near the periodic orbits O i for i = 1, 2.

The points (2, ±1) oe R are the local minimum or maximum for the function g and thus its matrix of first derivatives must also vanish at these points and the Hessian matrix of second derivatives must be positive semi-definite. The generic property for such a function is that the Hessian matrix for g at these points is positive definite. In the works [P5, P7], a more precise version of this was formulated: HYPOTHESIS 1.6.1. The function g satisfies the following conditions:

g(r, z) = g 0 for (r ≠ 2) 2 + (|z| ≠ 1) 2 Ø ' 2 0 (1.17)
where 0 < ' 0 < 1/4 is su ciently small. Moreover, we require that the Hessian matrices of second partial derivatives for g at the vanishing points (2, ±1) are positive definite. In addition, we require that g(r, z) is monotone increasing as a function of the distance  (r ≠ 2) 2 + (|z| ≠ 1) 2 from the points (2, ±1).

Many of the results in [P5, P7] require this generic hypotheses for their proofs, as it allows making estimates on the speed of ascent for the orbits of the Wilson flow near the periodic orbits.

The choices for the embeddings ‡ i : D i ae W, for i = 1, 2, as illustrated on Figures 1.7 and 1.8, are more wide-ranging and have a fundamental influence on the dynamics of the resulting Kuperberg flows on the quotient space K. We first impose a "normal form" condition on the insertions, which CHAPTER 1. A MINIMAL SET does not have significant impact on the dynamics, but allows a more straightforward formulation of the other properties of the insertion maps.

Let (r, ◊, z)

= ‡ i (x Õ ) oe D i for i = 1, 2, where x Õ = (r Õ , ◊ Õ , z Õ ) oe D i is a point in the domain of ‡ i . Let fi z (r, ◊, z) = (r, ◊, ≠2
) denote the projection of W along the z-coordinate. We assume that ‡ i restricted to the bottom face, ‡ i : L ≠ i ae W, has image transverse to the vertical fibers of fi z . This normal form can be achieved by an isotopy of a given embedding along the flow lines of the vector field W, so does not change the orbit structure of the resulting vector field on the plug K.

The above transversality assumption implies that fi

z ¶ ‡ i : L ≠ i ae W is a di eomorphism into the face ˆ≠ h W, with image denoted by D i µ ˆ≠ h W. Then let Ë i = (fi z ¶ ‡ i ) ≠1 : D i ae L ≠
i denote the inverse map, so we have:

Ë i (r, ◊, ≠2) = (r(Ë i (r, ◊, ≠2)), ◊(Ë i (r, ◊, ≠2)), ≠2) = (R i,r (◊), i,r (◊), ≠2) . (1.18)
We can then formalize in terms of the maps Ë i the assumptions on the insertion maps ‡ i . HYPOTHESIS 1.6.2 (Strong Radius Inequality). For i = 1, 2, assume that:

1. ‡ i : L ≠ i ae W is transverse to the fibers of fi z ; 2. r = r( ‡ i (r Õ , ◊ Õ , z Õ )) < r Õ , except for (2, ◊ i , z Õ ) and then z( ‡ i (2, ◊ i , z Õ )) = (≠1) i ; 3. i,r (◊) = ◊(Ë i (r, ◊, ≠2)) is an increasing function of ◊ for each fixed r; 4. R i,r (◊) = r(Ë i (r, ◊, ≠2)) has non-vanishing derivative for r = 2, except for the case of ◊ i defined by Ë i (2, ◊ i , ≠2) = (2, ◊ i , ≠2); 5.
For r su ciently close to 2, we require that the ◊ derivative of R i,r (◊) vanish at a unique point denoted by ◊(i, r), with ◊(i, 2) = ◊ i . Consequently, each surface L ≠ i is transverse to the coordinate vector fields ˆ/ˆ◊ and ˆ/ˆz on W. The illustration of the image of the curves {r Õ = 2} and {r Õ = 3} on Figure 1.8 suggests that these curves have "parabolic shape". We formulate this notion more precisely using the function Ë i (r, ◊, ≠2) defined by (1.18) and introduce the more general hypotheses they may satisfy. Recall that ' 0 > 0 was introduced in Hypothesis 1.6.1. HYPOTHESIS 1.6.3. For i = 1, 2, 2 AE r 0 AE 2 + ' 0 and ◊

i ≠ ' 0 AE ◊ AE ◊ i + ' 0 , assume that d d◊ i,r0 (◊) > 0 , d 2 d◊ 2 R i,r0 (◊) > 0 , d d◊ R i,r0 (◊ i ) = 0 . (1.19) where ◊ i satisfies Ë i (2, ◊ i , ≠2) = (2, ◊ i , ≠2
). Thus for 2 AE r 0 AE 2 + ' 0 , the graph of R i,r0 (◊) is convex upwards with vertex at ◊ = ◊ i . Hypothesis 1.6.3 implies that all of the level curves {r Õ = c}, for 2 AE c AE 2 + ' 0 , have parabolic shape, as the illustration in Figure 1.8 suggests.

We can now define what is called a generic Kuperberg flow in the work [P5].

DEFINITION 1.6.4. A Kuperberg flow t is generic if the Wilson flow W used in the construction of the vector field K satisfies Hypothesis 1.6.1 and the insertion maps ‡ i for i = 1, 2 used in the construction of K satisfies Hypotheses 1.6.2, and Hypotheses 1.6.3.

That is, the singularities for the vanishing of the vertical component g • ˆ/ˆz of the vector field W are of quadratic type and the insertion maps used to construct K yield quadratic radius functions near the special points.

Recall that the insertion maps for the variations from a Kuperberg flow as introduced in Section 1.5 are denoted by ‡ ' i : D i ae W, for i = 1, 2. It is assumed that these maps satisfy the

CHAPTER 2. TRUNKENNESS, AN ASYMPTOTIC INVARIANT FOR FLOWS

Recently, it was proved by Kudryavtseva for vector fields obtained by suspending an areapreserving di eomorphism of a surface [START_REF] Kudryavtseva | Conjugation invariants on the group of area-preserving di eomorphisms of the disk[END_REF] and for non-vanishing vector fields [START_REF] Kudryavtseva | Helicity is the only invariant of incompressible flows whose derivative is continuous in C 1 -topology[END_REF] and then by Enciso, Peralta-Salas and Torres de Lizaur [START_REF] Enciso | Helicity is the only integral invariant of volume-preserving transformations[END_REF] for arbitrary volume-preserving vector fields, that every invariant that is regular integral (in the sense that its Fréchet derivative is the integral of a continuous kernel) is a function of helicity. I will not state the precise statements. These results give a satisfactory explanation of why most constructions yield invariants that are functions of helicity for ergodic vector fields. However they do not rule out the existence of other invariants, but imply that such invariants cannot be too regular.

An example of such another invariant is the asymptotic crossing number considered by Freedman and He [START_REF] Freedman | Divergence-Free Fields: Energy and Asymptotic Crossing Number[END_REF]. The advantage is that it is not proportional to helicity, but the disadvantage is that it is hard to compute, even on simple examples.

In [P9] we consider a less well-known knot invariant called the trunk (see Definition 2.0.1 below). It was defined by Ozawa [START_REF] Ozawa | Waist and trunk of knots[END_REF], building on the concept of thin position that was introduced by Gabai [Gab87] for solving the R-conjecture. Less famous than the invariants previously studied in the context of vector fields, the trunk has the advantage that its definition relies on surfaces transverse to the considered knot, so that it is easy to transcript in the context of vector fields. The invariant depends on an invariant measure for the flow of the vector field that may or may not be a volume and is invariant under homeomorphisms that preserve this measure. Given a µ-preserving vector field X and a surface S, the geometric flux through S is the infinitesimal volume that crosses S in both directions (see Definition 2.0.4), we denote it by Flux(X, µ, S). Our invariant is a minimax of the geometric flux, where one minimizes over all height functions and maximizes over the levels of the considered height function.

To motivate our definition, I will start by defining the trunk of a knot. Let K be a knot, embedded in R 3 . The standard height function on R 3 is the function h z : R 3 ae R, (x, y, z) ' ae z. Every level h ≠1 z (t) is a 2-dimensional plane. A height function on R 3 is a function obtained by precomposing h z by an orientation preserving di eomorphism, that is, a function of the form h : R 3 ae R (x, y, z) ' ae h z ("(x, y, z)) for " an orientation preserving di eomorphism of R 3 . In particular, a height function is a function whose levels are smooth planes. A height function h is said to be in Morse position with respect to K if the restriction of h to K is a Morse function. In this case there are only finitely many points at which K is tangent to a level of h. We denote this set of functions by height Morse functions. DEFINITION 2.0.1. Assume that K is an embedded knot in R 3 that is in Morse position with respect to h z . The trunk of the curve K relatively to h

z is tk hz (k) := max toeR ˘{K fl h ≠1 z (t)}.
The trunk of a knot K is then defined by

) = Tks(X, µ).
Now for p a point in the ambient manifold and t > 0, denote by k X (p, t) the closed curve obtained by concatenating the orbit segment between p and " t X (p) and a short path between these two points. As for the asymptotic linking number, there is a system of short paths such that we obtain a simple closed curve for µ-almost all points and almost all times [START_REF] Vogel | On the asymptotic linking number[END_REF]. The next corollary states that the trunkenness is an asymptotic invariant in the sense of Arnold [START_REF] Arnold | Topological methods in hydrodynamics[END_REF] and its order is 1. COROLLARY 2.0.9. Assume that X is a µ-preserving vector field and that X is ergodic with respect to µ, then for µ-almost every p the limit

lim taeOE 1 t Tk(k X (p, t))
exists and is equal to Tks(X, µ).

Theorem 2.0.7 allows to compute the trunkenness of Seifert flows on S 3 , as explained in Section 2.1. These computations in turn show that the trunkenness is not dictated by helicity, even in the case of ergodic vector fields, thus contrasting with most previously known knot-theoretical constructions.

THEOREM 2.0.10. There is no function f such that for every ergodic volume-preserving vector field X on S 3 one has Tks(X, µ) = f (Hel(X, µ)).

Finally we address the question of what happens if for a non-singular vector field on S 3 there is a function that achieves the trunkenness, or in other words if the infimum in Definition 2.0.5 is a minimum. THEOREM 2.0.11. Let X be a non-singular vector field on S 3 preserving the measure µ and h a height function such that Tks(X, µ) = max toe[0,1] Flux(X, µ, h ≠1 (t))

Then X has an unknotted periodic orbit.

One of the main motivations for constructing topological invariants of a vector field X is to find lower bounds on the energies E p (X) := s |X| p dµ. Indeed since a topological invariant yields a time-independent invariant of the physical system, an energy bound in terms of a topological invariant will also be time-independent, although the energy may vary when the vector field is transported under (volume-preserving) di eomorphisms. Such energy bounds exist for the helicity and for the asymptotic crossing number. We do not know whether the trunkenness bounds the energy, a possible path towards finding an inequality between the trunk and the energy is to relate the trunk to the crossing number that is itself related to helicity.

Independence of helicity

As mentioned in the introduction, the helicity is a well-known invariant of vector fields up to volume-preserving di eomorphism. In this section, all vector fields are on the sphere S 3 and preserve a volume form, that we denote by . For X such a vector field, Cartan's formula implies that ÿ X is a closed 2-form and since the ambient manifold is simply connected it is exact. We may then write ÿ X = d-, forsome di erential 1-form. The helicity of X can be defined as As we recalled in the introduction, most known asymptotic invariants are in fact proportional to the helicity [START_REF] Arnold | The asymptotic Hopf invariant and its applications[END_REF][START_REF] Gambaudo | Signature asymptotique d'un champ de vecteurs en dimension 3[END_REF][START_REF] Baader | Asymptotic concordance invariants for ergodic vector fields[END_REF][START_REF] Baader | Asymptotic Vassiliev invariants for vector fields[END_REF]. The goal of this section it to prove that the trunkenness of a vector field is not a function of its helicity. In order to do so we compute the trunkenness and the helicity of a vector field that preserves the invariant tori of a Hopf fibration of S 3 .

Considering S 3 as the unit sphere {(z 1 , z 2 ) oe C 2 , |z 1 | 2 + |z 2 | 2 = 1}, the Seifert flow of slope (-, -) is the flow " -,-given by " t -,-(z 1 , z 2 ) := (z 1 e i2fi-t , z 2 e i2fi-t ), generated by the vector field X -,-. This flow preserves the standard volume form, that is, the volume form Haar associated to the Haar measure of S 3 . The flow has two distinctive periodic orbits corresponding to z 1 = 0 and z 2 = 0 that are trivial knots in S 3 . The tori |z 1 /z 2 | = r for 0 < r < OE are invariant and the flow on each one of them is the linear flow of slope -/-. If -/is rational, put -/-= p/q with p, q oe N coprime. Then every orbit of " -,-, di erent from the two trivial ones, is a torus knot of type T (p, q). The helicity of " -,-is equal to --. To compute it in the rational case (-, -) = (p, q) with p, q coprime, observe that all the orbits except two are periodic of period 1. The linking number of an arbitrary pair of such orbits is pq. Therefore the asymptotic linking number (also called asymptotic Hopf invariant) equals pq and, by Arnold's Theorem [START_REF] Arnold | The asymptotic Hopf invariant and its applications[END_REF], so does the helicity. For the general case of (-, -) not necessarily rational, it is enough to use the continuity of the helicity, since X -,can be approximated by a sequence of Seifert flows with rational slope.

PROPOSITION 2.1.1. The trunkenness of the Seifert flow "

-,-with respect to the standard volume form Haar is equal to 2 min(-, -).

Proof. Let us first prove Tks(X -,-, Haar ) AE 2-. For this it is enough to exhibit a height function h that yields tks h (X -,-, Haar ) = 2-. First define OE = (0, 1) and 0 = (0, ≠1) in S 3 µ C 2 and take the stereographic projection to identify

{(z 1 , z 2 ) oe C 2 , |z 1 | 2 + |z 2 | 2 = 1} ƒ R 3 fi {OE}.
Take now as h the standard height function h 0 of R 3 fi {OE}. The spheres are centered at 0 oe R 3 that corresponds to the point (0, ≠1) oe S 3 µ C 2 , hence the orbit z 1 = 0 intersects twice each level sphere h ≠1 0 (t). The middle sphere, S = h ≠1 0 (1/2), contains the other special orbit z 2 = 0 and is the only sphere that intersects all the orbits of " -,-. Then the function t ' ae For computing s S |ÿ X -,-Haar |, we remark that the 2-sphere S has the orbit (e i2fi-t , 0) as an equator, that the flow is positively transverse to the northern hemisphere and negatively transverse to the southern hemisphere. Then the integral s S |ÿ X -,-Haar | is equal to twice the flux of X -,-through any disk bounded by the curve (e i2fi-t , 0). Consider the flat disk D in S 3 bounded by (e i2fi-t , 0). The first return time to D is constant and equal to 1/-, so the flux For proving the converse inequality Tks(X -,-, Haar ) Ø 2 min(-, -), we approximate X -,-(in the C OE -topology) by a sequence (X pn/rn,qn/rn ) noeN , where p n , q n , r n are integer numbers. Theorem 2.0.8 yields Tks(X -,-, Haar ) = lim naeOE Tks(X pn/rn,qn/rn , Haar ). As the trunkenness is an order-1 invariant (which means that it is multiplied by ⁄ if the vector field is multiplied by ⁄), we only have to prove Tks(X p,q , Haar ) = 2 min(p, q) for p, q two coprime natural numbers.

Since every orbit of X p,q is periodic, we can consider a sequence (K n ) noeN of collections of periodic orbits whose induced normalized linear Dirac measures µ n converge to Haar . We take K n to be an n-component link all of whose components are torus knots T (p, q). Actually K n is a cabling with n strands on T (p, q), so by Zupan's theorem [START_REF] Zupan | A lower bound on the width of satellite knots[END_REF], the trunk of K n is 2n min(p, q). Since the period of each component of K n is 1, the total length of K n is n and we get Tks(X p,q , Haar ) = lim naeOE Tks(X p,q , µ n ) = lim naeOE 2 min(p, q) = 2 min(p, q).

Proof of Theorem 2.0.10. The previous computations show that for a Seifert flow X -,-on S 3 we have Hel(X -,-, Haar ) = -and Tks(X -,-, Haar ) = 2 min(-, -). There is no real function g such that min(-, -) = g(--), so there is no function g such that Tks(X -,-, Haar ) = g(Hel(X -,-, Haar )). However the Seifert flows are not ergodic with respect to Haar . Indeed, the foliation of S 3 by tori is invariant, so that it is easy to construct an invariant set with arbitrary measure. Still, a theorem of Katok [START_REF] Katok | Ergodic perturbations of degenerate integrable Hamiltonian systems[END_REF] states that the vector field X -,-can be perturbed (in the C 1 -topology) into an ergodic one. Starting from X 1,8 and X 2,4 and applying Katok's argument, we obtain two ergodic volume-preserving vector fields X Õ 1,8 and X Õ 2,4 . By continuity, their trunknesses are close to 2 and 4 respectively, while their helicities are close to 8. At the expense of multiplying the X Õ 1,8 and X Õ 2,4 by a constant, we can assume that their helicities are exactly 8. However their trunknesses are still close to 2 and 4, hence di erent.

It is also interesting to compare the formula Tks(X -,-, Haar ) = 2 min(-, -) with Kudryavtseva's and Encisco-Peralata-Salas-Torres de Lizaur's theorems: the function (-, -) ' ae 2 min(-, -) is continuous but not di erentiable, so that trunkenness is a continuous vector field invariant, but it is not integral regular in the sense of [START_REF] Kudryavtseva | Helicity is the only invariant of incompressible flows whose derivative is continuous in C 1 -topology[END_REF][START_REF] Enciso | Helicity is the only integral invariant of volume-preserving transformations[END_REF].

10

  Travaux de recherche (P9) The trunkenness of a volume-preserving vector field. En collaboration avec Pierre Dehornoy. À paraître dans Nonlinearity. (P10) Perspectives on Kuperberg flows. En collaboration avec Steve Hurder.

Figure 1

 1 Figure 1.1: A plug trapping a periodic orbit

Figure 1 . 2 :

 12 Figure 1.2: Vector field W v

Figure 1

 1 Figure 1.3: W-orbits on the cylinders {r = cst.} Define the closed subsets: R = {(2, ◊, z) | ≠1 AE z AE 1} [The Reeb Cylinder] A = {z = 0} [The Center Annulus]

Figure 1

 1 Figure 1.4: W-orbits on the cylinder {r = 2} and in W

  Figure 1.5, the construction continues with the choice in the annulus [1, 3] ◊ S 1 of two closed regions L i , for i = 1, 2, which are topological disks. Each region has boundary defined by two arcs: for i = 1, 2, -Õ i is the boundary contained in the interior of [1, 3] ◊ S 1 andi in the outer boundary contained in the circle {r = 3}, as depicted in Figure 1.6.

Figure 1 . 6 :

 16 Figure 1.6: The disks L 1 and L 2

  follow the W-trajectory and traverse the insertion from the bottom face to the top face. Since W is vertical near the boundary of W and horizontal at the two periodic orbits, we have that the arcs ‡ i (x ◊ [≠2, 2]) are vertical near the inserted curve ‡ i (-Õ i ) and horizontal at the intersection of the insertion with the periodic orbit O i . Thus, the embeddings of the surfaces ‡ i (L i ◊ {z}) make a half turn upon insertion, for each ≠2 AE z AE 2. The turning is clockwise for the bottom insertion i = 1 as illustrated in Figure1.7 and counter-clockwise for the upper insertion i = 2, which is illustrated in Figure1.9.

Figure 1 . 7 :

 17 Figure 1.7: The image of D 1 under ‡ 1

  Figure 1.11: The cylinder {r = 2} in " W

Figure 1 .

 1 Figure 1.13: The intersection of the orbit of p Õ (1; 1, 2) with the cylinder {r = r 2 }The intersection of the W-orbit of p Õ (1; 1, 2) with W Õ is a collection of W-arcs between transition points, contained in the cylinder {r = r 2 }. The second part of Proposition 1.3.1 means that the image under • of all these arcs is contained in S 1 and in the same order. Let us call these arcs A j for 1 AE j AE n for some finite n (n is finite since the W-orbit of p Õ (1; 1, 2) is finite, in Figure1.13 we have n = 5).Clearly the image under • of the arcs above does not completely fills the gap between p(1; 1, 2) and p(1; 1, 2), since it is not a connected set. To do so, we need to understand what happens

Figure 1 .1

 1 Figure 1.19: R Õ µ W

  1 and L ≠ 2 are subsets of the entry region of W, consider the two infinite propellers P " µ W generated by the curve " Õ and P ⁄ generated by ⁄ Õ in W. The set M 1 0 of points with level at most one consists of the image under • of the Reeb cylinder and the two infinite propellers P " and P ⁄ . Each of these propellers intersects infinitely many times the insertions, thus P Õ " = P " fl W Õ and P Õ ⁄ = P ⁄ fl W Õ have infinitely many notches each. In analogy with Figure 1.15, Figure 1.22 is a flattened diagram of M 1 0 . The horizontal band is the notched Reeb cylinder R Õ , whose lower boundary is O Õ 1 and whose upper boundary is O Õ 2 , union the boundaries of the notches. The two vertical bands correspond to P Õ " (the lower one) and P Õ ⁄ (the upper one), parts of their boundaries are contained in S 1 and S 2 respectively. The set M 1 0 1.3. THE MINIMAL SET UNDER THE GENERIC HYPOTHESES 33 in K is embedded di erently, since the two infinite propellers turn around the Reeb cylinder that is itself embedded as in Figure 1.20. I wont attempt to draw an image of this.

Figure 1 .

 1 Figure 1.22: Flattened M 1 0

Figure 1 .

 1 Figure 1.23: Curves in • (P Õ " ) fl E 1

Figure 1 .

 1 Figure 1.24: Flattened M 0

i

  are sketched in the right-hand side illustration in Figure1.26.Let me comment on some details of the regions in Figures 1.26

  (a) and (b). For the map " + i , i = 1, 2, the domain contains a neighborhood of the point Ê i . Flowing the domain U i containing points with r-coordinate equal to 2. Observe that the Radius Inequality implies that the maximum radius of points in Á U " + i is bigger than the maximum radius of points in U " + i . The first intersection of the W-orbits of points

Figure 1

 1 Figure 1.26: Domains and ranges for the maps {"+ 1 , " + 2 , " ≠ 1 , " ≠ 2 }

Figure 1 .

 1 Figure 1.27: A pseudo-orbit as generator of the homology of the shape approximation

Hel-

  • d-, and does not depends on the choice of the primitive -[AK98].

  |ÿ X -,-Haar | has a maximum for t = 1/2.

2. 1 .

 1 INDEPENDENCE OF HELICITY55multiplied by 1/-gives the total volume of S 3 , that is 1. Therefore Flux(X -,-, Haar , D) is equal toand we obtain tks h0 (X -,-, Haar ) = s S |ÿ X -,-Haar | = 2-. By symmetry, we then have Tks(X -,-, Haar ) AE 2 min(-, -).

  The transition points of an orbit of K are those CHAPTER 1. A MINIMAL SET points where the orbit intersects E 1 , E 2 , S 1 , S 2 or a boundary component ˆ≠ h K or ˆ+ h K. They are then either primary or secondary transition points, where x oe K is:-a primary entry point if x oe ˆ≠ h K or a primary exit point if x oe ˆ+ h K; -a secondary entry point if x oe E 1 fi E 2 or a secondary exit point if x oe S 1 fi S 2 . If a K-orbit contains no transition points, then it lifts under

  A knot is trivial if and only if its trunk equals 2. Indeed the embedding as the boundary of a vertical disk shows that the trunk is less than or equal to 2 and every embedding THEOREM 2.0.7. The trunkenness is a continuous functional on the space of normal currents.

	Tk(K) := One can define the trunk of a knot by fixing the height function and changing the embedding of min h height Morse function tk h (K) = min h height Morse function max toeR ˘{K fl h ≠1 (t)}. the knot in R 3 , see [Zup12]. n , µ n ) noeN is a sequence of measure-preserving vector fields such that (X n ) noeN converges to X in the C 0 -topology and (µ n ) noeN converges to µ in the weak-ú sense. Then we have naeOE EXAMPLE 2.0.2. COROLLARY 2.0.8. Suppose that (X lim Tks(X n , µ n

Remerciements

LIST OF FIGURES

Observe that the vertical lines are not continuous: the dotted parts correspond to the arcs that are not in the intersection of these two W-orbits with W Õ . In other words, in the left-hand side vertical line the continuous segments correspond (from top to bottom) to the arcs from p(1) to p(1; 1, 1), from p(1; 1, 1) to p(1; 1, 2), from p(1; 1, 2) to p(1; 1, 3) and in general from p(1; 1, ¸) to p(1; 1, ¸+ 1) for ¸Ø 1 and unbounded. The right-hand side vertical line is the backward orbit of p(1) that repeatedly intersects the second insertion.

Let me point out one aspect in this new set of flattened diagrams. In the plug K the left-hand side vertical continuous segments in Figure 1.15 correspond to arcs that accumulate on the horizontal segment • (O Õ 1 ), while the right-hand side vertical continuous segments accumulate on

The level 2 of S 1 is composed by all the finite W-orbits starting at the intersections of the two level 1 orbits with E 1 and E 2 . Thus by (the image under • of) a countable collection of finite W-orbits. Observe that since these are finite Wilson orbits, we just need to consider the starting Figure 1.16: Level 0, 1 and 2 of S 1 with gaps

Entropy of the minimal set

As mentioned in the introduction to this chapter, as a consequence of Katok's theorem on C 2 -flows on 3-manifolds [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for di eomorphisms[END_REF], the topological entropy of a Kuperberg flow is zero. Indeed, Katok's theorem gives an upper bound of the topological entropy by the growth of the number of periodic orbits with respect to the period. Hence any aperiodic flow has topological entropy zero. In [P5] we use G ú K , introduced in Definition 1.4.1, to give an explicit computation of the topological entropy. This computation revealed chaotic behavior for the flow, but such that it evolves at a very slow rate and thus it does not result in positive topological entropy. We proved that, under some extra hypotheses, such generic Kuperberg flows have positive slow entropy and that the rates of chaotic behavior grow at a precise subexponential rate, as discussed in Chapter 21 of [P5].

To make the discussion more precise, let me give the definitions. Define the entropy of the flow t using a variation of the Bowen formulation of topological entropy [START_REF] Bowen | Entropy for group endomorphisms and homogeneous spaces[END_REF] for a flow on a compact metric space (X, d X ). The definition we adopt is symmetric in the role of the time variable t. Two points p, q oe X are said to be (Ï

Then the topological entropy is defined by I recall now the definition of entropy for pseudoıgroups as introduced by Ghys, Langevin and Walzcak in [START_REF] Ghys | Entropie géométrique des feuilletages[END_REF]. Let (X, d X ) be a compact metric space and G

(1)

k } be a set of local homeomorphisms of X, with their inverses. Let G X denote the pseudogroup generated by the collection of maps G

(1) X . Let G ú X µ G X denote the pseudoıgroup generated by the compositions of maps in G

(1) X (as remarked after Definition 1.4.1, this might not be a pseudogroup). Let G

X be the collection of maps defined by the restrictions of compositions of at most n elements of G

(1)

X such that x, y are in the domain of Ï and d X (Ï(x), Ï(y)) > '. In particular, if the identity map is the only element of G (n) X which contains both x and y in its domain, then they are (G ú X , n, ')-separated if and only if

(1.15)

The property 0 < h GLW (G ú X ) < OE is independent of the choice of metric on X.

CHAPTER 1. A MINIMAL SET

The third entropy type invariant I want to introduce is the slow entropy of a map, first defined in the work of Katok and Thouvenot [START_REF] Katok | Slow entropy type invariants and smooth realization of commuting measure-preserving transformations[END_REF]. We adapted this idea for the action of a pseudoıgroup G ú X . For 0 < -< 1, define the --entropy of G ú X , or just the slow entropy, by

Analogously, one can define the slow entropy of a flow, by changing 1/T for 1/T -in (1.14).

The strategy in [P5] is to relate the flow entropy to the entropy of the pseudoıgroup G ú K . Estimating the entropy of G ú K is equivalent to understanding how the tree M 0 grows (see Remark 1.3.6). This is done by estimating the number of admissible words in the generators {Id, " + 1 , " ≠ 1 , " + 2 , " ≠ 2 , Â} and their inverses (I refer to Chapter 14 in [P5]). We prove that, for a generic Kuperberg flow, the number of words is a subexponential function of the length and thus the entropy of G ú K is zero. The computation suggests that there are exponentially many separated points, but the time that the flow or the pseudoıgroup take to separate them is very long. We thus decided to study the slow-entropy of the holonomy pseudoıgroup of the lamination that we denote by G M . I won't go into the details here, but this pseudoıgroup is generated by {Id, " + 1 , " + 2 , Â} and their inverses. It turned out that to obtain that h - GLW (G ú M ) > 0 for -= 1/2, we needed an extra hypothesis on the radius function defined by the insertions, that we baptized as slow-growth. Theorem 21.10 of [P5] states THEOREM 1.4.12. Let t be a generic Kuperberg flow. If the insertion maps ‡ j have "slow growth", then h

That the slow entropy of the lamination is positive implies (via a computation similar to the one carried out in Chapter 20 of [P5]), that the slow entropy of the flow t is positive. The words in the generators {Id, " + 1 , " + 2 , Â} used to prove Theorem 1.4.12 correspond to an exponentially growing family of loops that appear as generators of the first homology groups in the shape approximation used to prove Theorem 1.4.5. These two properties of look very much a like. I wonder if there is a general relation between the shape of a minimal set and the (slow) entropy of the flow. CONJECTURE 1.4.13. Assume that a flow on a closed 3-manifold M has an exceptional minimal set such that the rank of the first homology groups of a shape approximation grows exponentially, then the slow entropy of the flow is positive.

Positive entropy in a C OE -neighborhood of a generic Kuperberg plug

As remarked above, the topological entropy of a Kuperberg is zero, but the computations in [P5] revealed some chaotic behavior. In [P7] we study a C OE -perturbation of a generic Kuperberg flow and we prove the following theorem.

THEOREM 1.5.1. There exists a C OE 1-parameter family of plugs K

' for ' oe (≠1, a) for a > 0 such that:

1. The plug K 0 is a generic Kuperberg plug; 2. For ' < 0, the flow in the plug K ' has two periodic orbits that bound an invariant cylinder, every other orbit belongs to the wandering set and thus the flow has topological entropy zero; 1.5. VARIATIONS OF THE KUPERBERG PLUG 43 3. For ' > 0, the flow in K ' has positive topological entropy and an abundance of periodic orbits.

The proof of Theorem 1.5.1 for ' < 0 uses the same technical tools as developed in the previous works [START_REF] Kuperberg | A smooth counterexample to the Seifert conjecture[END_REF][START_REF] Kuperberg | Generalized counterexamples to the Seifert conjecture[END_REF][START_REF] Ghys | Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg)[END_REF][START_REF] Matsumoto | Kuperberg's C OE counterexample to the Seifert conjecture[END_REF] for the study of the dynamics of the Kuperberg plug K 0 . The proof for ' > 0 uses in a fundamental way the technique of relating the dynamics of a Kuperberg-like flow to the dynamics of its return map to an almost transverse rectangle, by introducing suitable pseudogroups as in Section 1.4.

I will not go into the details of the proof, but I explain the construction of the plugs K ' and give the main ideas to understand their dynamics. These plugs are constructed exactly as K in Section 1.2.1 except that conditions (K7) and (K8) on the embeddings ‡ ' i (in place of ‡ i ) are replaced by the following parametrized radius inequality. The embeddings ‡ ' i satisfy conditions (K1) to (K6) and are also required to satisfy two further conditions:

) and then r = 2 + '. Observe that for ' = 0, we recover the Radius Inequality of Kuperberg, one of the most fundamental concepts of Kuperberg's construction. Figure 1.28 represents the radius inequality for ' < 0, ' = 0, and ' > 0. Note that in the third illustration (c) the insertion has a vertical shift upwards. This is not required by conditions (K7') and (K8'), but it is used to prove the third part of Theorem 1.5.1. The case ' < 0

In the case ' < 0, the self-insertions do not penetrate far enough as to trap the periodic orbits. Condition (K8') implies that the radius always grows at secondary entry points (without any exceptions). For these plugs Proposition 1.3.1 is still valid. As a corollary we obtain that the K ' -orbits containing the arcs

in analogy with the definition of W Õ (1.5). As stated in Theorem 1.5.1, these two orbits are the only two periodic orbits of the flow in K ' . The proof of this last assertion is basically the same as the proof of the aperiodicity of K = K 0 .

CHAPTER 1. A MINIMAL SET

What I want to point out about this case is that it gives another way of understanding the minimal set of the original Kuperberg plug K = K 0 . In fact, the two periodic orbits of K ' for ' < 0 bound an invariant cylinder C ' that is obtained by flowing inside K ' the notched Reeb cylinder R Õ ' = R fl W Õ ' . Indeed, repeating the analysis in Section 1.3.2 we first point out that Proposition 1.3.4 is valid and we can consider as level 0 the set • ' (R Õ ' ) (where • ' is defined as • ). This set intersects each entrance of an insertion along a curve contained in the region {r > 2} and thus each curve generates a finite propeller that is at level 1 (see Figure 1.18 for an illustration of a finite propeller in W).

As ' increases to zero, the length of these two level 1 propellers grows and when ' = 0 the propellers become infinite. Hence for ' small enough, these two propellers at level 1 intersect the insertions and thus give birth to propellers at level 2, etc. So we can build the cylinder C ' as we built M 0 in Section 1.3.2. The main di erence is that for C ' the process stops: the level 1 propellers are finite and thus intersect the insertions a finite number of times and this repeats at any level, and the maximum level attained in C ' is upper bounded by a number that depends only on '. The set C

' is thus the union of the notched Reeb cylinder • ' (R Õ ' ) and a finite number of finite propellers.

A flattened illustration of C

' is basically the same as Figure 1.24, except that the two vertical level one propellers • (P Õ " ) and • (P Õ ⁄ ) are finite propellers, implying that the process described by Figures 1.22 and 1.24 stops. The length of the two level 1 propellers increases as ' increases to zero, thus the complexity of the flattened diagram increases. We thus obtain an approximation of M 0 by embedded cylinders in R 3 .

The case ' > 0

The Radius Inequality in this case is illustrated in Figure 1.28 (c). Along the vertical line {r = 2} the r Õ -coordinate is bounded below by 2 ≠ ', while the vertical line {r = 3} does not intersects the insertion. This implies that there are values (between 2 + ' and 3), let me call one of them a, of the radius such that along the vertical line {r = a} the r Õ -coordinate is bounded below by a. In [P7] we assume (for simplicity) that there is a unique such value, denoted by r ' (see Lemma 1.6.5 and Hypothesis 1.6.6 in the following section for a precise statement). Thus for points with r > r ' , we have that the radius coordinate grows at secondary entry points. This implies that Proposition 1.3.1 is valid when the hypothesis r > 2 is replaced by r > r ' . We can thus repeat the construction of the maps " + 1 , " + 2 , " ≠ 1 and " ≠ 2 in Section 1.4 and understand their action on points with radius coordinate bigger than r ' in the rectangle R 0 (for R 0 as in Section 1.4).

In order to prove Theorem 1.5.1 for ' > 0 we consider words in the generators {Id, " + 1 , ' } and their inverses, for ' as in (1.11). The proof then follows in two steps:

1. find a set V 0 µ R 0 for which there exists ¸(') oe N such that for every k Ø ¸(') and

proving that the invariant set of the maps Ï

k , for each k Ø ¸('), is an invariant set of the flow and that the return times to these sets are bounded.

The first point guarantees the existence of a countable number of horseshoe like dynamics in the pseudoıgroup generated by {Id, " + 1 , ' }. The second one says that these horseshoes are embedded in the flow and the fact that the return times are bounded, allows to conclude that the topological entropy of the flow is positive. The vertical shift in Figure 1.28 is used to prove this second point.

We do not understand how these horseshoes change as ' ae 0 and disappear in the plug K 0 . One thing I understand is that the return times to the invariant sets grow as ' decreases, but the 1.6. GENERIC HYPOTHESES modified conditions (K7') and (K8'). The illustrations of the radius inequality in Figure 1.28 again suggest that the images of the curves {r Õ = c} are of "quadratic type". For ' < 0 there is no need to make any further assumptions in the construction in order to obtain Theorem 1.5.1. In contrast, for ' > 0 we need the analog of the generic hypotheses in Definition 1.6.4 and the existence of the radius r ' as discussed in Section 1.5 and below. We again assume the insertion maps ‡ ' i : L ≠ i ae W are transverse to the fibers of the projection map fi z : W ae ˆ≠ h W along the z-coordinate. Then we can define the inverse map

and express the inverse map in polar coordinates as:

Then the level curves r Õ = c pictured in Figure 1.28 are given by the maps

We note a straightforward consequence of the Parametrized Radius Inequality (K8'). Recall that ◊ i is the radian coordinate specified in (K8') such that for

We then add an additional assumption on the insertion maps ‡ ' i for i = 1, 2 which specifies the qualitative behavior of the radius function for r Ø r ' . HYPOTHESIS 1.6.6. If r ' is the smallest

The conclusion of Hypothesis 1.6.6 is implied by the Radius Inequality for the case ' = 0, but does not follow from the condition (K8') when ' > 0.

We can now formulate the analog of Hypothesis 1.6.2, which imposes uniform conditions on the derivatives of the maps Ë ' i , for the plugs K ' with ' > 0. Recall that 0 < ' 0 < 1/4 was specified in Hypothesis 1.6.1 and we assume that 0 < ' < ' 0 . HYPOTHESIS 1.6.7 (Strong Radius Inequality). For i = 1, 2, assume that:

1.

) and then r = 2 + '; 3. ' i,r (◊) is an increasing function of ◊ for each fixed r; 4. For 2 ≠ ' 0 AE r AE 2 + ' 0 and i = 1, 2, assume that R ' i,r (◊) has non-vanishing derivative, except when ◊ = ◊ i as defined by

For r su ciently close to 2 + ', we require that the ◊ derivative of R ' i,r (◊) vanishes at a unique point denoted by ◊(i, r).

Note that Hypotheses 1.6.6 and 1.6.7 combined imply that r ' is the unique value of 2+' < r

We can then formulate the analog of Hypothesis 1.6.3. HYPOTHESIS 1.6.8. For 2 ≠ ' 0 AE r 0 AE 2 + ' 0 and ◊

where

Finally, we have the definition of the generic ' t flows of the plugs K ' , as introduced in Section 1.5 and studied in [P7]. DEFINITION 1.6.9. The flow ' t is generic if the Wilson flow W used in the construction of the vector field K ' satisfies Hypothesis 1.6.1 and the insertion maps ‡ ' i for i = 1, 2 used in the construction of K ' satisfies Hypotheses 1.6.7, and Hypotheses 1.6.8.

Chapter 2

Trunkenness, an asymptotic invariant for flows

This chapter covers the results of my work with P. Dehornoy [P9]. The problem we address is the construction of new invariants of volume-preserving vector fields on S 3 , or on compact domains of R 3 , up to volume-preserving di eomorphisms. This problem is motivated by at least two physical situations. First if v is the velocity field of a time-dependent ideal fluid satisfying the Euler equations (ideal hydrodynamics) then its vorticity field curl v is transported by the flow of v [Hel1858]. Second if B is the magnetic field of an incompressible plasma (ideal magnetodynamics), then B turns out to be transported by the velocity field as long as the latter does not develop singularities [START_REF] Woltjer | A Theorem on Force-free magnetic Fields[END_REF]. In these contexts, invariants of curl v or B up to volume-preserving di eomorphisms yield timeindependent invariants of the system.

Not so many such invariants exist. The first one was discovered by W. Thomson [Tho1867]: if the considered field has a periodic orbit or a periodic tube, then its knot type is an invariant (this remark led to the development of knot theory by P. G. Tait [Tait1877]). However it may not be easy to find periodic orbits and even then such an invariant only takes a small part of the field into account.

The main known invariant is called helicity. It is defined by the formula Hel(v) = s v • u, where u = curl ≠1 (v) is an arbitrary vector-potential of v. It was discovered by Woltjer, Moreau and Mo att [START_REF] Woltjer | A Theorem on Force-free magnetic Fields[END_REF][START_REF] Moreau | Constantes d'un îlot tourbillonnaire en fluide parfait barotrope[END_REF][START_REF] Moffatt | The degree of knottedness of tangle vortex lines[END_REF]. Helicity is easy to compute or to approximate since it is enough to exhibit a vector-potential of the considered vector field, to take the scalar product and to integrate. The connection with knot theory was sketched by Mo att [START_REF] Moffatt | The degree of knottedness of tangle vortex lines[END_REF] and deepened by Arnold [START_REF] Arnold | The asymptotic Hopf invariant and its applications[END_REF] as follows. Denote by k X (p, t) a loop starting at the point p, tangent to the vector field X for a time t and closed by an arbitrary segment of bounded length. Denote by Lk the linking number of loops. Arnold showed that for almost every p 1 , p 2 , the limit lim t1,t2aeOE

)) exists (see also [START_REF] Vogel | On the asymptotic linking number[END_REF] for a corrected proof). Moreover, if X is ergodic the limit coincides almost everywhere with Hel(X) (for a non-ergodic vector field, one has to average the previous limit to obtain the helicity).

The idea of considering knot invariants of long pieces of orbits of the vector field was pursued by Gambaudo and Ghys [START_REF] Gambaudo | Signature asymptotique d'un champ de vecteurs en dimension 3[END_REF] who considered Ê-signatures of knots, Baader [START_REF] Baader | Asymptotic concordance invariants for ergodic vector fields[END_REF] who considered linear saddle invariants, and Baader and Marché [START_REF] Baader | Asymptotic Vassiliev invariants for vector fields[END_REF] who considered Vassiliev's finite type invariants. In every case, it is shown that lim

exists, where V is the considered invariant and n a suitable exponent called the order of the asymptotic invariant. However all these constructions have the drawback that they do not yield any new invariant for ergodic vector fields, since the obtained limits are all functions of the helicity.

in Morse position of the trivial knot has to intersect some horizontal plane in at least two points. Conversely, if the trunk of a knot is equal to 2, then it admits an embedding that intersects every horizontal plane in at most two points. The union of the segments that connect these pairs of points is a disk bounded by the knot, implying that the knot is trivial. EXAMPLE 2.0.3. For p, q in N, the torus knot T (p, q) can be realized as the closure of a braid with q strands, yielding Tk(T (p, q)) AE 2q. By symmetry one also gets Tk(T (p, q)) AE 2p. Actually, one can prove Tk(T (p, q)) = 2 min(p, q), see Remark 1.2 in [START_REF] Ozawa | Waist and trunk of knots[END_REF]. We use Definition 2.0.1 to define the trunkenness of a vector field with respect to an invariant measure. The main question then concerns the analog of the number of intersection points of a surface with a curve when the curve is replaced by a vector field. A natural answer is the geometric flux. If X is a vector field that preserves a measure µ given by a volume element , one can then consider the 2-form ÿ X . For S a piece of oriented surface that is positively transverse to X, the integral s S ÿ X computes the instantaneous volume that crosses S. In other words, by Fubini Theorem we have µ(" [0,t] (S)) = ( s S ÿ X )•t. On the other hand if S is negatively transverse to X we have µ(" [0,t] (S)) = ≠( s S ÿ X ) • t. Therefore in this case, for any surface S, the instantaneous volume crossing S is given by s S |ÿ X |. Now if the measure µ is not given by integrating a volume form one cannot consider the above integral, but the quantity µ(" [0,t] (S)) still makes sense for any piece of surface S. DEFINITION 2.0.4. For X a vector field that preserves a measure µ and a surface S, the geometric flux of (X, µ) through S is Flux(X, µ, S) := lim

This definition generalizes the number of intersection points of a knot with a surface. Indeed one can see an embedding k of a knot K as a vector field with a particular invariant measure in the following way: on K, consider a non-singular vector field X k that is tangent to k at every point and denote by " t k the induced flow. Since k is closed, " k is T k -periodic for some T k > 0. The Dirac linear measure associated to X k is defined by

where A is a measurable set and x an arbitrary point on k. The measure µ k is X k -invariant and has total mass T k . In this setting, for S a surface that intersects k in finitely many points, a point p in the set k fl S has µ k -measure zero and thus cannot be detected by the measure. But by definition of µ k the set "

CHAPTER 2. TRUNKENNESS, AN ASYMPTOTIC INVARIANT FOR FLOWS

As µ k is concentrated on k, we thus have ˘{k fl S} = lim

so the geometric flux indeed generalizes the number of intersection points.

We now mimic for vector fields the definition of the trunk of a knot. In order to have a welldefined maximum, in what follows we assume the vector fields are on a compact domain D 3 µ R 3 or on the 3-sphere S 3 = R 3 fi OE. In the later case, we define the standard height function

The levels h ≠1 0 (0) and h ≠1 0 (1) consist of the points (0, 0, 0) and OE respectively and every other level h ≠1 0 (t) is a 2-dimensional sphere centered at the origin. A height function on S 3 is then a function obtained by precomposing by an orientation preserving di eomorphism " of S 3 , that is, a function of the form h : S 3 ae [0, 1], (x, y, z) ' ae h 0 ("(x, y, z)). DEFINITION 2.0.5. Let X be a vector field whose flow preserves a measure µ on a compact domain of R 3 or on S 3 and h a height function. We set

The trunkenness of (X, µ) is defined as

Note that we can only consider an infimum instead of a minimum as in the case of knots. In Theorem 2.0.11 we study the implications of having a height function achieving the trunkenness.

If the invariant measure µ is given by the integration of a volume form , we get the alternative definitions From Definition 2.0.5 it is straightforward that the trunkenness of a vector field is invariant under di eomorphisms that preserve the measure µ. More is true, the trunkenness is invariant under homeomorphisms that preserve µ. THEOREM 2.0.6. Assume that X 1 and X 2 are vector fields on S 3 or on a compact domain of R 3 that preserve a probability measure µ and that there is a µ-preserving homeomorphism f that conjugates the flows of X 1 and X 2 . Then we have Tks(X 1 , µ) = Tks(X 2 , µ).

What we do in [P9] is to prove several properties of this new invariant. The first one is a continuity result, that is more easily stated in terms of currents, but we also provide corollaries that do not rely on this vocabulary. Given a vector field X and a measure µ, we can define a normal 1-current C (X,µ) : forany di erential 1-form C (X,µ) (-) = s -(X)dµ, where the integral is taken on the ambient space. We consider the space of normal currents endowed with the mass topology (see [START_REF] Morgan | Geometric measure theory. A beginner's guide[END_REF]).

Dans ce mémoire je présente des résultats reliés à l'étude des flots en dimension 3.