According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no. The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.

According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no. The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.

Thesis (Ph.D.) - University of St Andrews, January 2007.

In this work we introduce the notion of Galois foliations on P2 , defined as those folations whose Gauss applications restricted to a Zariski open subset is a Galois covering. We also present some examples and acriterium for identifying such foliations. / En el presente trabajo introducimos la nocion de foliaciones Galois sobre P2C, definidas como aquellas cuya aplicacion de Gauss restringida aun abierto Zariski es un recubrimiento Galois. Asimismo, presentamo salgunos ejemplos y un criterio para identicar este tipo de foliaciones.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 1981 / Bibliography: leaves 102-103. / by Javier Bracho. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics

Ma, Man Shun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 75-76). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Basic notions in Riemannian geometry --- p.10 / Chapter 2.1 --- Basic manifold theory --- p.11 / Chapter 2.2 --- "Connection, curvature" --- p.19 / Chapter 2.3 --- Submanifold theory --- p.29 / Chapter 3 --- Basic facts in symplectic and complex geometry --- p.33 / Chapter 3.1 --- "Symplectic manifolds, Lagrangian submanifolds" --- p.34 / Chapter 3.2 --- Kahler and Calabi-Yau manifolds --- p.39 / Chapter 3.3 --- Calibration --- p.49 / Chapter 4 --- Mean curvature flow --- p.52 / Chapter 4.1 --- Basic equations in Lagrangian immersions --- p.53 / Chapter 4.2 --- Evolution equation for --- p.57 / Chapter 4.3 --- Evolution equations for H and θ --- p.62 / Chapter 5 --- Lagrangian angle of a foliation --- p.67 / Chapter 5.1 --- "Proof of equation (5.1), (5.2)" --- p.68 / Chapter 5.2 --- Main theorem --- p.70 / Chapter 5.3 --- Examples of invariant solution --- p.73 / Bibliography --- p.75

Soit f un germe de fonction holomorphe dans deux variables qui s'annule à l'origine. L'ensemble zéro de cette fonction définit un germe de courbe analytique. Bien que la classification topologique d'un tel germe est bien connue depuis les travaux de Zariski, la classification analytique est encore largement ouverte. En 2012, Hefez et Hernandes ont résolu le cas irréductible et ont annoncé le cas de deux components. En 2015, Genzmer et Paul ont résolu le cas des fonctions topologiquement quasi-homogènes. L'objectif principal de cette thèse est d'étudier la première classe topologique de fonctions non quasi-homogènes. Dans le deuxième chapitre, nous décrivons l'espace local des modules des feuillages de cette classe et nous donnons une famille universelle de formes normales analytiques. Dans le même chapitre, nous prouvons l'unicité globale de ces formes normales. Dans le troisième chapitre, nous étudions l'espace des modules de courbes, qui est l'espace des modules des feuillages à une équivalence analytique des séparatrices associées près. En particulier, nous présentons un algorithme pour calculer sa dimension générique. Le quatrième chapitre présente une autre famille universelle de formes normales analytiques, qui est globalement unique aussi. En effet, il n'ya pas de modèle canonique pour la distribution de l'ensemble des paramètres sur les branches. Ainsi, avec cette famille, nous pouvons voir que la famille précédente n'est pas la seule et qu'il est possible de construire des formes normales en considérant une autre distribution des paramètres. Enfin, pour la globalisation, nous discutons dans le cinquième chapitre une stratégie basée sur la théorie géométrique des invariants et nous expliquons pourquoi elle ne fonctionne pas jusqu'à présent. / Let f be a germ of holomorphic function in two variables which vanishes at the origin. The zero set of this function defines a germ of analytic curve. Although the topological classification of such a germ is well known since the work of Zariski, the analytical classification is still widely open. In 2012, Hefez and Hernandes solved the irreducible case and announced the two components case. In 2015, Genzmer and Paul solved the case of topologically quasi-homogeneous functions. The main purpose of this thesis is to study the first topological class of non quasi-homogeneous functions. In chapter 2, we describe the local moduli space of the foliations in this class and give a universal family of analytic normal forms. In the same chapter, we prove the global uniqueness of these normal forms. In chapter 3, we study the moduli space of curves which is the moduli space of foliations up to the analytic equivalence of the associated separatrices. In particular, we present an algorithm to compute its generic dimension. Chapter 4 presents another universal family of analytic normal forms which is globally unique as well. Indeed, there is no canonical model for the distribution of the set of parameters on the branches. So, with this family, we can see that the previous family is not the only one and that it is possible to construct normal forms by considering another distribution of the parameters. Finally, concerning the globalization, we discuss in chapter 5 a strategy based on geometric invariant theory and explain why it does not work so far.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

ARAUJO, H. R. M. Sobre invariantes topológicos de folheações holomorfas com singularidade isolada. 2017. 62 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-25T20:23:00Z No. of bitstreams: 1 2017_dis_hrmaraujo.pdf: 556971 bytes, checksum: 9f274c4a5c917004f3b67a3fc72c5547 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Conferi a Dissertação de HAMILTON REGIS MENEZES DE ARAUJO, e constatei apenas dois erros na formatação no trabalho que dever ser alterados pelo autor: 1- Epígrafe ( a estrutura desse elemento deve ser a que se segue, com alinhamento à direita: "O sucesso é ir de fracasso em fracasso sem perder o entusiasmo." (Winston Churchill) 2- Títulos das seções (os títulos das seções que se encontram no sumário e ao longo do texto estão incorretos. As normas da ABNT recomendam que apenas a primeira letra do título das seções esteja em maiúscula, com exceção de nomes próprios. Ex.: 2.2 Índice de um Campo em uma Singularidade Isolada deve ser alterado para: 2.2 Índice de um campo em uma singularidade isolada Atenciosamente, on 2017-05-26T16:04:16Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-29T13:43:41Z No. of bitstreams: 1 2017_dis_hrmaraujo.pdf: 556572 bytes, checksum: cae2a014846c47e96be936dd25bbd9da (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-05-29T14:07:05Z (GMT) No. of bitstreams: 1 2017_dis_hrmaraujo.pdf: 556572 bytes, checksum: cae2a014846c47e96be936dd25bbd9da (MD5) / Made available in DSpace on 2017-05-29T14:07:05Z (GMT). No. of bitstreams: 1 2017_dis_hrmaraujo.pdf: 556572 bytes, checksum: cae2a014846c47e96be936dd25bbd9da (MD5) Previous issue date: 2017-05-19 / Considering the foliation induced by a complex holomorph vector field, we will look for topological invariants in the neighborhood of a singular point. At first, the Milnor Number of a vector field becomes important, in the sense that this number is topological invariant. In another discussion, we will emphasize vector fields in dimension two, in which case the leaves, whose foliation is induced by the field, will be integral curves of a 1-form. In this sense, we will deal with Desingularization, that is, after a finite number of processes, which we will call Blow-ups or explosions, we will turn the initial foliation into a foliation whose singularities are all simple. Finally, the Desingularization process of a field will give us tools that make it possible to relate the data obtained in this process to the objects treated throughout the work, with this we will present other topological invariants of foliations. / Considerando a folheação induzida por um campo vetorial complexo holomorfo, buscaremos exibir invariantes topológicos na vizinhança de um ponto singular. Num primeiro momento, ganha importância o Número de Milnor de um campo vetorial, no sentido desse número ser invariante topológico. Em outra discussão, daremos ênfase a campos vetoriais em dimensão dois, nesse caso, as folhas, cuja folheação é induzida pelo campo, serão curvas integrais de uma 1-forma. Nesse sentido, trataremos de Desingularização, ou seja, após um número finito de processos, que chamaremos de Blow-ups, ou explosões, transformaremos a folheação inicial em uma folheação cujas singularidades são todas simples. Por fim, o processo de Desingularização de um campo nos dará ferramentas que possibilitam relacionar os dados obtidos nesse processo com os objetos tratados ao longo de todo o trabalho, diante disto apresentaremos outros invariantes topológicos de folheações.