Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we consider two sufficient conditions for the global injectivity of local diffeomorphisms X : R2 → R2 of class C1. The first is based on the spectrum of X, in this case it is enough to consider X differentiable, and the second is known as Palais-Smale Condition. In fact, these conditions ensure the triviality of the foliations in R2 induced by the coordinated functions of X and this guarantees the global injectivity of the map X. Besides discussing the proofs of this results, we exhibit a collection of examples showing that such conditions provide different classes of globally injective maps. / Neste trabalho consideramos duas condições suficientes para que um difeomorfismo local X : R2 → R2, de classe C1, seja globalmente injetivo. A primeira baseada no espectro da aplicação X, neste caso basta considerar X diferenciável, e a segunda é a condição de Palais-Smale. De fato, tais condições garantem a trivialidade das folheações em R2 induzidas pelas funções coordenadas de X e isto garante a injetividade global da aplicação X. Além de apresentar as demonstrações destes resultados, exibimos uma coleção de exemplos que permitem concluir que tais condições estabelecem classes distintas de aplicações globalmente injetivas. / Mestre em Matemática

Made available in DSpace on 2015-05-14T13:21:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 422678 bytes, checksum: a7a607df8d67afa93aa6137919ecb1f5 (MD5) Previous issue date: 2011-08-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The present work discusses a study of extactic curves in the projective plane, providing a method that guarantees the existence of first integrals for certain vector fields. To achieve this goal, this study covers the following topics: vector fields, first integrals (with the main result presented in Jouanolou's Theorem), holomorphic foliations (in particular, foliations on the projective plane) and algebraic solutions (where the main result is the well-known theorem of Darboux, which guarantees the existence of rational first integrals for algebraic foliations on the projective plane). / O presente trabalho aborda um estudo das curvas estáticas no plano projetivo, proporcionando um método que garante a existência de integrais primeiras para certos campos vetorias. Para atingir tal objetivo, o presente estudo abrange os seguintes tópicos: Campos Vetoriais, Integrais Primeiras (tendo como principal resultado apresentado o Teorema de Jouanolou), Folheações Holomorfas (em particular, folheações no plano projetivo) e as Soluções Algébricas (onde o principal resultado é o conhecido teorema de Darboux, que garante a existência de integrais primeiras racionais para folheações algébricas no plano projetivo).

Embedded in the study of Hilbert's innitesimal problem is the question of existence and number of limit cycles of linear perturbations of Hamiltonian fields. Since there is available a classication of real quadratic fields with center in R2, we can match them with complex fields in C2 that induce a foliation in P2. Our objective is to classify the foliations in P2 induced by the elds obtained by said classication of quadratic fields with center which are elliptic brations, that is, the ones with level curves of genus one. / En el estudio del problema infinitesimal de Hilbert se encuentra inmersa la tarea de analizar la existencia y de acotar el número de ciclos límite de una perturbación lineal de campos hamiltonianos. Como existe una clasificación de campos cuadráticos reales con centro en R2, podemos asociar campos complejos en C2 que inducen una foliación en P2. El objetivo de este trabajo es clasificar aquellas foliaciones en P2 inducidas por estos campos cuadráticos que sean fibraciones elípticas, es decir, aquellas cuyas curvas de nivel sean de género uno.

We present (without proof) a version of Bott theorem for compact complex orbifolds with isolated singularities. Then we deduce some important consequences of this theorem, and nally we give some applications to holomorphic foliations on weighted projective spaces. / Presentamos (sin demostración) una versión del teorema de Bott para un orbifold complejo compacto y con singularidades aisladas. A continuación deducimos algunas consecuencias importantes de este teorema, y finalmente daremos algunas aplicaciones para foliaciones holomorfas en espacios proyectivos ponderados.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T15:40:44Z No. of bitstreams: 1 arturafonsoguedesrossini.pdf: 414354 bytes, checksum: 7de0172af679cf245fd9b870000235c4 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T18:55:21Z (GMT) No. of bitstreams: 1 arturafonsoguedesrossini.pdf: 414354 bytes, checksum: 7de0172af679cf245fd9b870000235c4 (MD5) / Made available in DSpace on 2017-05-29T18:55:21Z (GMT). No. of bitstreams: 1 arturafonsoguedesrossini.pdf: 414354 bytes, checksum: 7de0172af679cf245fd9b870000235c4 (MD5) Previous issue date: 2011-12-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Uma folheação algébrica do plano projetivo sobre um corpo k pode ser dada tanto por um campo de vetores como por uma 1-forma em P2, já que dimensão um e codimensão um são a mesma noção visto que a dimensão de P2 é igual a 2. Então surge uma pergunta natural: Como se relacionam os campos vetoriais e as 1-formas em P2? Veremos que uma 1-forma ω e um campo de vetores X definem a mesma folheação do plano projetivo quando ω(p)(X(p)) = 0 para todo ponto p ∈P2. Uma segunda questão é a existência de curvas algébricas invariantes por uma folheação de P2. Originalmente, Poincaré formulou o seguinte problema: É possível limitar o grau de uma curva algébrica invariante por um campo de vetores em termos do grau do campo de vetores? A resposta para este problema é negativa, como podemos ver no Exemplo 3.18. Entretanto adicionando-se algumas hipóteses sobre tal curva invariante este problema pode possuir resposta positiva. No caso em que tal curva invariante é suave, mostra-se que o grau da curva é no máximo igual ao grau do campo vetorial mais um. Se uma curva invariante não for suave, mostra se que ainda é possível limitar o grau desta curva em termos do grau da folheação e da regularidade do seu conjunto de singularidades. / An algebraic foliation of the projective plane over a field k can be given either by a vector field or a 1-form in P2, as dimension one and codimension one are the same notion since dim(P2) = 2. Then a natural question arises: How do vector fields and 1-forms in P2 relate? We will see that an 1-form ω is related with a vector field X belonging to the kernel of ω, that is, ω and X define the same foliation of the projective plane when ω(p)(X(p)) = 0 for all points p ∈P2. A second question concerns about the existence of algebraic curves that are invariant by a foliation of P2. Originally, Poincaré formulated the following problem: Is it possible to bound the degree of an invariant curve under a vector field in terms of the degree of the field? The problem has a negative answer, but by adding some hypothesis it can be reformulated in order to have a positive answer. If we assume that this invariant curve is smooth, we show that the degree of the curve is at most the degree of the vector field plus one. If an invariant curve is not smooth, we show that its degree can be limited in terms of regularity of its set of singularities.

Orientador: Paulo Regis Caron Ruffino / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T22:03:50Z (GMT). No. of bitstreams: 1 GonzalesGargate_IvanItalo_D.pdf: 724522 bytes, checksum: bb313a00b360e7bea2a2411b759c7389 (MD5) Previous issue date: 2012 / Resumo: Nesta tese, estudamos um princípio de médias em equações diferenciais estocásticas sobre variedades folheadas com folhas compactas. Começaremos introduzindo o princípio de médias sobre equações diferenciais ordinárias reais. A título de comparação vamos rever conceitos básicos de variedade simplética com a finalidade de comparar/estender os resultados obtidos por Xue-Mei Li sobre um princípio de médias para um sistema Hamiltoniano estocástico completamente integrável. Nosso principal resultado é generalizar estas idéias para o caso de uma variedade M = (-a; a)n x N, onde N é uma variedade compacta sem bordo. Em particular mostraremos nossos resultados para o caso que a folheação é gerada por uma submersão de M sobre Rn. Finalmente apresentamos alguns exemplos / Abstract: In this thesis, we study the averaging principle for stochastic differential equations on foliated manifolds with compact leaves. We begin by introducing the averaging principle over real ordinary differential equations. For comparison we will review basic concepts of symplectic manifold in order to compare/extend the results obtained by Xue-Mei Li about a averaging principle for a completely integrable stochastic Hamiltonian system. Our main result is to generalize these ideas to the case of a manifold M = (-a; a)n x N, where N is a compact manifold without boundary. In particular our results show for the case that foliation is generated by an submersion of M over Rn. Finally we present some examples / Doutorado / Matematica / Doutor em Matemática

In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u â—¦ dW

La thèse présente une description géométrique d’un germe de famille générique déployant un champ de vecteurs réel analytique avec un foyer faible à l’origine et son complexifié : le feuilletage holomorphe singulier associé. On montre que deux germes de telles familles sont orbitalement analytiquement équivalents si et seulement si les germes de familles de difféomorphismes déployant la complexification de leurs fonctions de retour de Poincaré sont conjuguées par une conjugaison analytique réelle. Le “caractère réel” de la famille correspond à sa Z2-équivariance dans R^4, et cela s’exprime comme l’invariance du plan réel sous le flot du système laquelle, à son tour, entraîne que l’expansion asymptotique de la fonction de Poincaré est réelle quand le paramètre est réel. Le pullback du plan réel après éclatement par la projection monoidal standard intersecte le feuilletage en une bande de Möbius réelle. La technique d’éclatement des singularités permet aussi de donner une réponse à la question de la “réalisation” d’un germe de famille déployant un germe de difféomorphisme avec un point fixe de multiplicateur égal à −1 et de codimension un comme application de semi-monodromie d’une famille générique déployant un foyer faible d’ordre un. Afin d’étudier l’espace des orbites de l’application de Poincaré, nous utilisons le point de vue de Glutsyuk, puisque la dynamique est linéarisable auprès des points singuliers : pour les valeurs réels du paramètre, notre démarche, classique, utilise une méthode géométrique, soit un changement de coordonée (coordonée “déroulante”) dans lequel la dynamique devient beaucoup plus simple. Mais le prix à payer est que la géométrie locale du plan complexe ambiante devient une surface de Riemann, sur laquelle deux notions de translation sont définies. Après avoir pris le quotient par le relèvement de la dynamique nous obtenons l’espace des orbites, ce qui s’avère être l’union de trois tores complexes plus les points singuliers (l’espace résultant est non-Hausdorff). Les translations, le caractère réel de l’application de Poincaré et le fait que cette application est un carré relient les différentes composantes du “module de Glutsyuk”. Cette propriété implique donc le fait qu’une seule composante de l’invariant Glutsyuk est indépendante. / The thesis gives a geometric description for the germ of the singular holomorphic foliation associated with the complexification of a germ of generic analytic family unfolding a real analytic vector field with a weak focus at the origin. We show that two such germs of families are orbitally analytically equivalent if and only if the germs of families of diffeomorphisms unfolding the complexified Poincaré map of the singularities are conjugate by a real analytic conjugacy. The Z2-equivariance of the family of real vector fields in R^4 is called the “real character” of the system. It is expressed by the invariance of the real plane under the flow of the system which, in turn, carries the real asymptotic expansion of the Poincaré map when the parameter is real. After blowing up the singularity, the pullback of the real plane by the standard monoidal map intersects the foliation in a real Möbius strip. The blow up technique allows to “realize” a germ of generic family unfolding a germ of diffeomorphism of codimension one and multiplier −1 at the origin as the semi-monodromy of a generic family unfolding an order one weak focus. In order to study the orbit space of the Poincaré map, we perform a trade-off between geometry and dynamics under the Glutsyuk point of view (where the dynamics is linearizable near the singular points): in the resulting “unwrapping coordinate” the dynamics becomes much simpler, but the price we pay is that the local geometry of the ambient complex plane turns into a much more involved Riemann surface. Over the latter, two notions of translations are defined. After taking the quotient by the lifted dynamics we get the orbit space, which turns out to be the union of three complex tori and the singular points (this space is non- Hausdorff). The Glutsyuk invariant is then defined over annular-like regions on the tori. The translations, the real character and the fact that the Poincaré map is the square of the semi-monodromy map, relate the different components of the Glutsyuk modulus. That property yields only one independent component of the Glutsyuk invariant.

En 1979, Trotman a démontré que les stratifications réelles lisses qui satisfont la condition de (a)-régularité sont précisément celles pour lesquelles la transversalité aux strates des applications est une condition stable dans la topologie forte. C'était un résultat surprenant puisque la (t)-régularité semblait être plus appropriée pour la stabilité de la transversalité, une erreur qui a été faite dans plusieurs articles avant que ce résultat soit montré par Trotman. Notre premier résultat est un analogue au résultat de Trotman pour la topologie faible.Il y a une dizaine d'années Trotman a demandé si le même résultat est valable pour les stratifications analytiques complexes. Dans ce travail on démontre un analogue du résultat de Trotman dans le cas complexe, en utilisant la notion de variété de Oka introduite par Forstneric et on montre que la conjecture n'est pas vraie en général en donnant des contre-exemples.Dans sa thèse, Trotman a formulé une conjecture pour généraliser son résultat pour les stratifications (a_f)-régulières de Thom. Dans une tentative de résolution de cette conjecture on a observé que la transversalité par rapport à un feuilletage est une condition stable, cependant ce n'est pas une condition générique. Donc, en voulant imiter la preuve de Trotman on ne pourra pas obtenir cette généralisation. Néanmoins, on donne ici une preuve de cette conjecture. Ce résultat peut être résumé en disant que les (a_f)-défauts dans une stratification peuvent être détectés en perturbant les applications transverses au feuilletage induit par f. Certaines techniques pour détecter (a_f)-défauts sont aussi données vers la fin. / Trotman in 1979 proved that real smooth stratifications which satisfy the condition of $(a)$-regularity are precisely those stratifications for which transversality to the strata of smooth mappings is a stable condition in the strong topology. This was a surprising result since $(t)$-regularity seemed to be more appropriate for stability of transversality, a mistake that was made in several articles before this result of Trotman. Our first result is an analogue of this result of Trotman for the weak topology.Trotman asked more than ten years ago whether a similar result holds for complex analytic stratifications. We will give an analogue of Trotman's result in the complex setting using Forstneriv c's notion of Oka manifolds and show that the result is not true in general by giving counterexamples.In his Ph.D. thesis Trotman conjectured a generalization of his result for Thom $(a_f)$-regular stratifications. In an attempt to prove this conjecture we noticed that while transversality to a foliation is a stable condition, it is not generic in general. Thus, mimicking the proof of the result of Trotman would not suffice to obtain this generalization. Nevertheless, we will present a proof of this conjecture in this work. This result can be summarized by saying that Thom $(a_f)$-faults in a stratification can be detected by perturbation of maps transverse to the foliation induced by $f$. Some other techniques of detecting $(a_f)$-faults are also given towards the end.

Nesta tese estudamos os sistemas integráveis definidos em superfícies compactas possuindo uma integral primeira que é uma função Morse-Bott a valores em R. Estes sistemas são aqui chamados de sistemas integráveis Morse-Bott. Classificamos as curvas fechadas e oitos associados a pontos de selas imersos em superfícies compactas. Essa classificação é aplicada ao estudo das folheações Morse-Bott em superfícies e nos permite definir um invariante topológico completo para a classificação topológica global destas folheações. Como uma aplicação desse estudo obtemos a classificação dos sistemas Morse-Bott assim como a classificação topológica das funções Morse-Bott em superfícies compactas e orientáveis. Demonstramos ainda um teorema da realização baseado em duas transformações e numa folheação geradora. Para o caso das funções Morse-Bott também obtivemos um teorema de realização. Finalmente, investigamos a generalização de alguns dos resultados anteriores para sistemas definidos em superfícies não orientáveis. / In this thesis we study integrable systems on compact surfaces with a first integral as a Morse-Bott function with target R. These systems are called here integrable Morse-Bott systems. Initially we present the classification of closed curves and eights associated to saddle points on compact surfaces. This classification is applied to the study of Morse- Bott foliations on surfaces allowing us to define a complete topological invariant for the global topological classification of these foliations. Then as an application of this study we obtain the classification of integrable Morse-Bott systems as well as the topological classification of Morse-Bott functions on compact and orientable surfaces. We also prove a realization theorem based on two transformation and a generating foliation (the foliation on the sphere with two centers). In the case of Morse-Bott functions we also obtain a realization theorem. Finally we investigate generalizations of previous results for systems defined on non-orientable surfaces.