Dans cette thèse, nous nous intéressons à diverses propriétés algébriques des groupes d'homéomorphismes et de difféomorphismes de variétés. On appelle fragmentation la possibilité d'écrire un homéomorphisme en tant que composé d'homéomorphismes supportés dans des boules. Tout d'abord, nous étudions la longueur des commutateurs sur le groupe des homéomorphismes du tore et de l'anneau, ainsi que la norme de fragmentation, qui associe à tout homéomorphisme le nombre minimal de facteurs nécessaires pour écrire cet homéomorphisme en tant que composé d'homéomorphismes supportés dans des boules. Dans une deuxième partie de la thèse, nous abordons una autre propriété algébrique des groupes d'homéomorphismes et de difféomorphismes : la distorsion. Celle-ci est reliée de manière surprenante à des propriétés de fragmentation des homéomorphismes. / In this thesis, we are interested in various algebraic properties of groups of homeomorphisms and diffeomorphisms of manifolds. We call fragmentation the possibility to write a homeomorphism as a composition of homeomorphisms supported in balls. First, we study the commutator length on the group of homeomorphisms of the torus and of the annulus, as well as the fragmentation norm, which associates to any homeomorphism the minimal number of factors necessary to write this homeomorphism as a composition of homeomorphisms supported in balls. In a second part of this thesis, we deal with another algebraic property of homeomorphism and diffeomorphism groups: the distortion. This last notion is surprisingly related to fragmentation properties of homeomorphisms.

Dans quelle mesure peut-on lire les propriétés dynamiques (quand le temps tend vers l’infini) d’un système sur des simulations numériques ? Pour tenter de répondre à cette question, on étudie dans cette thèse un modèle rendant compte de ce qui se passe lorsqu’on calcule numériquement les orbites d’un système à temps discret f (par exemple un homéomorphisme). L’ordinateur travaillant à précision numérique finie, il va remplacer f par une discrétisation spatiale de f, notée f_N (où l’ordre de la discrétisation N rend compte de la précision numérique). On s’intéresse en particulier au comportement dynamique des applications finies f_N pour un système f générique et pour l’ordre N tendant vers l’infini, où générique sera à prendre dans le sens de Baire (principalement parmi des ensembles d’homéomorphismes ou de C^1-difféomorphismes). La première partie de cette thèse est consacrée à l’étude de la dynamique des discrétisations f_N lorsque f est un homéomorphisme conservatif/dissipatif générique d’une variété compacte. L’étude montre qu’il est illusoire de vouloir retrouver la dynamique du système de départ f à partir de celle d’une seule discrétisation f_N : la dynamique de f_N dépend fortement de l’ordre N. Pour détecter certaines dynamiques de f il faut considérer l’ensemble des discrétisations f_N, lorsque N parcourt N.La seconde partie traite du cas linéaire, qui joue un rôle important dans l’étude du cas des C^1-difféomorphismes génériques, abordée dans la troisième partie de cette thèse. Sous ces hypothèses, on obtient des résultats similaires à ceux établis dans la première partie, bien que plus faibles et de preuves plus difficiles. / How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations ? To try to answer this question, we study inthis thesis a model reflecting what happens when the orbits of a discrete time system f (for example an homeomorphism) are computed numerically. The computer working in finite numerical precision, it will replace f by a spacial discretization of f, denotedby f_N (where the order N of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or C^1-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations f_N, when f is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of f from that of a single discretization f_N : its dynamics strongly depends on the order N. To detect some dynamical features of f we have to consider all thediscretizations f_N when N goes through N.The second part deals with the linear case, which plays an important role in the study of C^1-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part,though weaker and harder to prove.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Mostraremos que é possível aproximar um difeomorfismo simplético com derivada contínua por um difeomorfismo simplético, infinitamente diferenciáveis, sobre uma variedade simplética compacta. Além disso, provamos o Teorema de Darboux e Moser.

L’objectif à moyen terme de ce travail est d’explorer quelques formulations des problèmes d’identification de forme et de reconnaissance de surface à partir de mesures ponctuelles. Ces problèmes ont plusieurs applications importantes dans les domaines de l’imagerie médicale, de la biométrie, de la sécurité des accès automatiques et dans l’identification de structures cohérentes lagrangiennes en mécanique des fluides. Par exemple, le problème d’identification des différentes caractéristiques de la main droite ou du visage d’une population à l’autre ou le suivi d’une chirurgie à partir des données générées par un numériseur. L’objectif de ce mémoire est de préparer le terrain en passant en revue les différents outils mathématiques disponibles pour appréhender la géométrie comme variable d’optimisation ou d’identification. Pour l’identification des surfaces, on explore l’utilisation de fonctions distance ou distance orientée, et d’ensembles de niveau comme chez S. Osher et R. Fedkiw ; pour la comparaison de surfaces, on présente les constructions des métriques de Courant par A. M. Micheletti en 1972 et le point de vue de R. Azencott et A. Trouvé en 1995 qui consistent à générer des déformations d’une surface de référence via une famille de difféomorphismes. L’accent est mis sur les fondations mathématiques sous-jacentes que l’on a essayé de clarifier lorsque nécessaire, et, le cas échéant, sur l’exploration d’autres avenues. / The mid-term objective of this work is to explore some formulations of shape identification and surface recognition problems from point measurements. Those problems have important applications in medical imaging, biometrics, security of the automatic access, and in the identification of Lagrangian Coherent Structures in Fluid Mechanics. For instance, the problem of identifying the different characteristics of the right hand or the face from a population to another or the follow-up after surgery from data generated by a scanner. The objective of this mémoire is to prepare the ground by reviewing the different mathematical tools available to apprehend the geometry as an identification or optimization variable. For surface identification it explores the use of distance functions, oriented distance functions, and level sets as in S. Osher and R. Fedkiw ; for surface recognition it emphasizes the construction of Courant metrics by A. M. Micheletti in 1972 and the point of view of R. Azencott and A. Trouvé in 1995 which consists in generating deformations of a reference surface via a family of diffeomorphisms. The accent will be put on the underlying mathematical foundations that it will attempt to clarify as necessary, and, if need be, on exploring new avenues.

As famílias Anosov foram introduzidas por P. Arnoux e A. Fisher, motivados por generalizar a noção de difeomorfismo de Anosov. A grosso modo, as famílias Anosov são sequências de difeomorfismos (fi)i&#8712Z definidos em uma sequencia de variedades Riemannianas compactas (Mi)i&#8712Z, em que fi: Mi ->Mi+1 para todo i &#8712 Z, tal que a composição fi+no· · ·ofi, para n >=1, tem comportamento assintoticamente hiperbólico. Esta noção é conhecida como um sistema dinâmico não-estacionário ou um sistema dinâmico não-autônomo. Sejam M a união disjunta de cada Mi, para i &#8712 Z, e Fm(M) o conjunto consistente das famílias de difeomorfismos (fi)i&#8712Z de classe Cm definidos na sequência (Mi)i&#8712Z. O propósito principal deste trabalho é mostrar algumas propriedades das famílias Anosov. Em particular, mostraremos que o conjunto destas famílias é aberto em Fm(M), em que Fm(M) é munido da topologia forte (ou topologia Whitney); a estabilidade estrutural de certa classe de famílias Anosov, considerando conjugações topológicas uniformes; e várias versões para os Teoremas de variedades estáveis e instáveis. Os resultados que serão apresentados aqui generalizam alguns outros resultados obtidos em Sistemas Dinâmicos Aleatórios, os quais serão mencionados ao longo do trabalho. Além do anterior, será introduzida a entropia topológica para elementos em Fm(M) e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua em Fm(M) munido da topologia forte. Porém, ela é descontínua em cada elemento de Fm(M) munido da topologia produto. Também apresentaremos um resultado que pode ser uma ferramenta de muita utilidade no estudo da continuidade da entropia topológica de difeomorfismos definidos em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao longo desta pesquisa e que serão analisados em um trabalho futuro. / Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the notion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms (fi)i&#8712Z dened on a sequence of compact Riemannian manifolds (Mi)i&#8712Z, where fi: Mi -> Mi+1 for all i &#8712 Z, such that the composition fi+n o · · · o fi, for n >=1, has asymptotically hyperbolic behavior. This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system. Let M be the disjoint union of each Mi, for each i &#8712 Z, and Fm(M) the set consisting of families of Cm-dieomorphisms (fi)i&#8712Z dened on the sequence (Mi)i&#8712Z. The main goal of this work is to explore some properties of Anosov families. In particular, we will show that the set consisting of these families is open in Fm(M), where Fm(M) is endowed with the strong topology (or Whitney topology); the structural stability of a certain class of Anosov families, considering uniform topological conjugacies; and some versions of stable and unstable manifold theorems. The results that will be presented here generalize some results obtained in Random Dynamical Systems, which will be mentioned throughout the work. In addition to the above mentioned theorems, the topological entropy for elements in Fm(M) will be introduced, and we will show some of its properties. We will prove that this entropy is continuous on Fm(M) endowed with strong topology. However, it is discontinuous at each element of Fm(M) endowed with the product topology. We will also present a result that can be a very useful tool in the study of the continuity of the topological entropy of dieomorphisms dened on compact manifolds. We will nish the work by giving a list of problems that have arisen throughout this research and that will be analyzed in a future work.

As famílias Anosov foram introduzidas por P. Arnoux e A. Fisher, motivados por generalizar a noção de difeomorfismo de Anosov. A grosso modo, as famílias Anosov são sequências de difeomorfismos (fi)i&#8712Z definidos em uma sequencia de variedades Riemannianas compactas (Mi)i&#8712Z, em que fi: Mi ->Mi+1 para todo i &#8712 Z, tal que a composição fi+no· · ·ofi, para n >=1, tem comportamento assintoticamente hiperbólico. Esta noção é conhecida como um sistema dinâmico não-estacionário ou um sistema dinâmico não-autônomo. Sejam M a união disjunta de cada Mi, para i &#8712 Z, e Fm(M) o conjunto consistente das famílias de difeomorfismos (fi)i&#8712Z de classe Cm definidos na sequência (Mi)i&#8712Z. O propósito principal deste trabalho é mostrar algumas propriedades das famílias Anosov. Em particular, mostraremos que o conjunto destas famílias é aberto em Fm(M), em que Fm(M) é munido da topologia forte (ou topologia Whitney); a estabilidade estrutural de certa classe de famílias Anosov, considerando conjugações topológicas uniformes; e várias versões para os Teoremas de variedades estáveis e instáveis. Os resultados que serão apresentados aqui generalizam alguns outros resultados obtidos em Sistemas Dinâmicos Aleatórios, os quais serão mencionados ao longo do trabalho. Além do anterior, será introduzida a entropia topológica para elementos em Fm(M) e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua em Fm(M) munido da topologia forte. Porém, ela é descontínua em cada elemento de Fm(M) munido da topologia produto. Também apresentaremos um resultado que pode ser uma ferramenta de muita utilidade no estudo da continuidade da entropia topológica de difeomorfismos definidos em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao longo desta pesquisa e que serão analisados em um trabalho futuro. / Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the notion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms (fi)i&#8712Z dened on a sequence of compact Riemannian manifolds (Mi)i&#8712Z, where fi: Mi -> Mi+1 for all i &#8712 Z, such that the composition fi+n o · · · o fi, for n >=1, has asymptotically hyperbolic behavior. This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system. Let M be the disjoint union of each Mi, for each i &#8712 Z, and Fm(M) the set consisting of families of Cm-dieomorphisms (fi)i&#8712Z dened on the sequence (Mi)i&#8712Z. The main goal of this work is to explore some properties of Anosov families. In particular, we will show that the set consisting of these families is open in Fm(M), where Fm(M) is endowed with the strong topology (or Whitney topology); the structural stability of a certain class of Anosov families, considering uniform topological conjugacies; and some versions of stable and unstable manifold theorems. The results that will be presented here generalize some results obtained in Random Dynamical Systems, which will be mentioned throughout the work. In addition to the above mentioned theorems, the topological entropy for elements in Fm(M) will be introduced, and we will show some of its properties. We will prove that this entropy is continuous on Fm(M) endowed with strong topology. However, it is discontinuous at each element of Fm(M) endowed with the product topology. We will also present a result that can be a very useful tool in the study of the continuity of the topological entropy of dieomorphisms dened on compact manifolds. We will nish the work by giving a list of problems that have arisen throughout this research and that will be analyzed in a future work.

L’objectif à moyen terme de ce travail est d’explorer quelques formulations des problèmes d’identification de forme et de reconnaissance de surface à partir de mesures ponctuelles. Ces problèmes ont plusieurs applications importantes dans les domaines de l’imagerie médicale, de la biométrie, de la sécurité des accès automatiques et dans l’identification de structures cohérentes lagrangiennes en mécanique des fluides. Par exemple, le problème d’identification des différentes caractéristiques de la main droite ou du visage d’une population à l’autre ou le suivi d’une chirurgie à partir des données générées par un numériseur. L’objectif de ce mémoire est de préparer le terrain en passant en revue les différents outils mathématiques disponibles pour appréhender la géométrie comme variable d’optimisation ou d’identification. Pour l’identification des surfaces, on explore l’utilisation de fonctions distance ou distance orientée, et d’ensembles de niveau comme chez S. Osher et R. Fedkiw ; pour la comparaison de surfaces, on présente les constructions des métriques de Courant par A. M. Micheletti en 1972 et le point de vue de R. Azencott et A. Trouvé en 1995 qui consistent à générer des déformations d’une surface de référence via une famille de difféomorphismes. L’accent est mis sur les fondations mathématiques sous-jacentes que l’on a essayé de clarifier lorsque nécessaire, et, le cas échéant, sur l’exploration d’autres avenues. / The mid-term objective of this work is to explore some formulations of shape identification and surface recognition problems from point measurements. Those problems have important applications in medical imaging, biometrics, security of the automatic access, and in the identification of Lagrangian Coherent Structures in Fluid Mechanics. For instance, the problem of identifying the different characteristics of the right hand or the face from a population to another or the follow-up after surgery from data generated by a scanner. The objective of this mémoire is to prepare the ground by reviewing the different mathematical tools available to apprehend the geometry as an identification or optimization variable. For surface identification it explores the use of distance functions, oriented distance functions, and level sets as in S. Osher and R. Fedkiw ; for surface recognition it emphasizes the construction of Courant metrics by A. M. Micheletti in 1972 and the point of view of R. Azencott and A. Trouvé in 1995 which consists in generating deformations of a reference surface via a family of diffeomorphisms. The accent will be put on the underlying mathematical foundations that it will attempt to clarify as necessary, and, if need be, on exploring new avenues.

Au début du XXème siècle, Poincaré puis Birkhoff ont été amenés, lors de leur recherche sur le problème restreint des trois corps, à étudier les courbes invariantes par une transformation d’une surface préservant l’aire. Cinquante ans plus tard, les théorèmes KAM démontrent la persistance de courbes invariantes après perturbation en topologie de classe k plus grande ou égale à trois. On peut alors se demander ce que devient ce résultat en topologie de classe moins élevée. Par ailleurs, l’étude des dynamiques C1-génériques connaît de nombreux développements, grâce notamment au Connecting Lemma. Par exemple, Bonatti et Crovisier on démontré qu’un difféomorphisme C1-générique d’une telle surface possède un ensemble dense de points dont l’orbite sort de tout compact. Ces deux résultats permettent de penser qu’un difféomorphisme C1-générique d’une surface n’admet pas de courbes fermées simples invariantes. C’est ce que nous démontrons dans ce travail. On obtient assez facilement, en utilisant le Connecting Lemma ainsi que les propriétés topologiques de l’anneau, qu’un difféomorphisme C1-générique de l’anneau possède des points périodiques sur toute courbe fermée simple invariante. Cela se généralise à une surface quelconque en utilisant une famille dénombrable d’anneau constituant une base de voisinages d’une courbe fermée simple quelconque. La construction d’une telle famille d’anneaux est le principal résultat du premier chapitre. Il s’agit alors de supprimer les points périodiques sur les courbes invariantes. Dans un premier temps, nous nous inspirerons d’un argument qu’Herman utilise dans le cadre de courbes invariantes par les twists de l’anneau pour montrer que tous les points périodiques ne peuvent être hyperboliques. Ensuite, nous définissons une propriété, la propriété G, qui si elle est vérifiée par un difféomorphisme symplectique et l’un de ses points périodiques elliptiques, empêche que ce point périodique appartienne à une courbe invariante. En montrant que cette propriété est vérifiée par un difféomorphisme C1-générique et tous ses points périodiques elliptiques, nous obtenons le résultat souhaité. Dans le quatrième chapitre, nous nous employons à définir de façon rigoureuse la notion de fonction génératrice qui est l’outil classique pour perturber des difféomorphismes symplectiques / Poincaré and Birkhoff were led, during their research on the restricted problem of three bodies, to study invariant curves under an area preserving map of a surface. Fifty years later, theorems KAM show the persistance of invariant curves in topology Ck with k greater or equal to three. What becomes this result in topology class lower. Moreover, the study of C1-generic dynamics knows many developments particulary through the Connecting Lemma. For example, Bonatti and Crovisier showed a C1-generic symplectic diffeomorphism of a compact surface is transitive. What they have adapted with M.-C. Arnaud to a non compact surface : a C1-generic symplectic diffeomorphism of a non compact surface has a dense set of points whose orbit leaves every compacts. These two results suggest a such application has not an invariant simple closed curve. The proof of this result is the aim of this work. We obtain, using the Connecting Lemma, a C1-generic symplectic diffeomorphism has periodic points on all the invariant curves. Then, deleting the periodic points from the invariant curves is the challenge. At first, we use an argument that Herman used in the context of curves invariant by a twist of annulus, to show that all periodic points cannot be hyperbolic. Then, we define a property, the property G, which, if it is verified by a symplectic diffeomorphism and one of its periodic elliptic points, prevents this periodic point belongs to an invariant curve. By showing that property is verified by a C1-generic symplectic diffeomorphism, we obtain the desired result. In the fourth chapter, we explain how to pertube a symplectic diffeomorphism with generating functions