Given a Geometric Lorenz Flow X on \'R POT. n+2\' of class \'C POT. k+1\'; by definition there exists a Poincaré map \'P IND. X\' of class \'C POT. k+1\'; often so-called Lorenz-type map [ABS83]. The main purpose in this dissertation is to show that under certain conditions the Lorenz-type map \'P IND.X\' can be associate to it a one-dimensional transformation \'f IND. X\' of class \'C POT. k\' (defined on an interval). This association is so-called the reduction transformation R; so we have \'RP IND. X\' = \'f IND. X\'. This association would allow us to study the dynamical properties for the original flow using techniques of one-dimensional dynamics of class \'C POT. k\' / Dado um Fluxo Geométrico de Lorenz X em \'R POT. n+2\' de classe \'C POT. k+1\'; por definição, existe uma aplicação de Poincaré \'P IND. X\'\' de classe \'C POT. k+1\'; frequentemente chamado aplicação do tipo Lorenz [ABS83]. O objetivo principal desta tese é mostrar que, sob certas condições a aplicação do tipo Lorenz \'P IND. X\' pode ser associado a ele uma transformação unidimensional \'f IND. X\' de classe \'C POT. k\' (definida em um intervalo). Esta associação é chamada de transformação de redução R; assim temos que , \'RP IND. X\' = \'f IND. X\'. Esta associação nos permitiria estudar as propriedades dinâmicas do fluxo original utilizando técnicas da dinâmica unidimensional de classe \'C POT. k\'

A geometria de teias dedica-se ao estudo de invariantes locais para uma determinada configuração de folheações. Uma d-teia é uma coleção de folheações que estão em posição geral. Desta forma, uma d-teia plana, definida em \'R POT.2\' ou \'C POT.2\', nada mais é que uma família de d folheações por curvas. Apresentamos neste trabalho os principais conceitos da teoria clássica de teias, iniciada por W. Blaschke por volta de 1930, bem como uma abordagem atual utilizada no estudo de teias planas. São abordados dois tipos de problemas importantes na teoria: os problemas de linearização e de algebrização de teias. Provamos um resultado clássico no que concerne ao problema de linearização, e um resultado de algebrização de teias empregando métodos desenvolvidos mais recentemente / Web geometry is devoted to the study of local invariants of a certain configuration of foliations. A d-web is a collection of foliations in general position. Therefore, a d-web defined in \'R POT. 2\' or \'C POT. 2\' is just a family of d foliations by curves. We present in this work the main concepts of classical theory of webs, initiated by W. Blaschke around 1930, as well as newer methods used in the study of plane webs. We approach two important types of problems in the theory: problems of linearization and that of algebrization of webs. We prove a classical result concerning the linearization problem, and a result of algebrization of webs using recently developed methods

Considérons une variété régulière analytique M sur le corps réel ou complexe, un faisceau d'idéaux J défini sur M, un diviseur à croisement normaux simples E et une distribution singulière involutive Θ tangent à E.L'objectif principal de ce travail est d'obtenir une résolution des singularités du faisceau d'idéaux J qui préserve certaines ``bonnes" propriétés de la distribution singulière Θ. Plus précisément, la propriété de R-monomialité : l'existence d'intégrales premières monomiales. Ce problème est naturel dans le contexte où on doit étudier l'interaction d'une variété et d'un feuilletage et, donc, est aussi reliée au problème de la monomilisation des applications et de résolution ``quasi-lisse" des familles d'idéaux.- Le premier résultat donne une résolution globale si le faisceau d'idéaux J est invariant par la distribution singulière;- Le deuxième résultat donne une résolution globale si la distribution singulière Θ est de dimension 1 ;- Le troisième résultat donne une uniformisation locale si la distribution singulière Θ est de dimension 2.On présente aussi deux utilisations des résultats précédents. La première application concerne la résolution des singularités en famille analytique, soit pour une famille d'idéaux, soit pour une famille de champs de vecteurs. Pour la deuxième, on applique les résultats à un problème de système dynamique, motivé par une question de Mattei. / Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and Θ an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some ``good" properties of the singular distribution Θ. More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the ``interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of ``quasi-smooth" resolution of families of ideal sheaves.- The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution Θ;- The second result is a global resolution if the the singular distribution Θ has leaf dimension 1;- The third result is a local uniformization if the the singular distribution Θ has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.

Στο πρώτο κεφάλαιο της εργασίας, παρουσιάζεται το πιο απλό παράδειγμα εμφύλλωσης και στη συνέχεια δίνεται ο ορισμός μιας εμφυλλωμένης πολλαπλότητας, υπό δύο διαφορετικές σκοπιές. Ο ορισμός συμπληρώνεται με τον σχολιασμό της τοπολογίας των φύλλων της εμφύλλωσης, δίνοντας το τοπολογικό πλαίσιο της πολλαπλότητας για τον ορισμό του κανονικού εμφυλλωμένου άτλαντα. Η εισαγωγή στη Θεωρία Εμφυλλώσεων ολοκληρώνεται με μία σειρά παραδειγμάτων εμφυλλώσεων, με επικεντρωμένο το ενδιαφέρον στην εμφύλλωση του Reeb και στην προσανατολισμένη εμφύλλωση του Seifert. Στο δεύτερο κεφάλαιο, συνδέεται η έννοια της γεωμετρικής ολοκληρωσιμότητας με την Θεωρία των Εμφυλλώσεων, μέσω του κλασικού θεωρήματος του Frobenius. Τα φύλλα της εμφύλλωσης του χώρου των φάσεων αποτελούν το γεωμετρικό πρότυπο επίλυσης δυναμικών συστημάτων, ως πρώτα ολοκληρώματα. Το κλασικό θεώρημα του Frobenius έδωσε τις αναγκαίες και ικανές συνθήκες ώστε η θεωρούμενη κατανομή να αποτελεί τον εφαπτόμενο χώρο της εμφύλλωσης. Το θεώρημα Frobenius δίνεται και αποδεικνύεται με πέντε ισοδύναμες εκδοχές. Μία από αυτές είναι η αλγεβρική εκδοχή, όπου τα πρώτα ολοκληρώματα καθορίζονται από τους γεννήτορες του ιδεώδους της εξωτερικής άλγεβρας, επιλύoντας τις εξισώσεις Pfaff. Οι παραγόμενες μορφές μέσω της εξωτερικής διαφόρισης των γεννητόρων του ιδεώδους, στην περίπτωση που ικανοποιούν τη συνθήκη ολοκληρωσιμότητας, συγκροτούν στο module των διαφορικών μορφών το διαφορικό ιδεώδες. Ακόμα, γίνεται αναφορά στο Λήμμα του Poincaré, που δίνει τις προϋποθέσεις για την ύπαρξη πρώτων ολοκληρωμάτων, στην περίπτωση απλά συνεκτικών πολλαπλοτήτων, και στην εύρεση ολοκληρωτικού παράγοντα. Στο τρίτο και τελευταίο κεφάλαιο, ως εφαρμογή στη Θεωρία Εμφυλλώσεων, αποδεικνύεται η ύπαρξη φύλλων μέσα στο σύνολο προσβασιμότητας, που καθορίζεται από το εκάστοτε σύστημα ελέγχου. Πρόκειται για το θεώρημα που δόθηκε τη δεκαετία του 70 από τον Sussmann. Ορίζοντας τη Lie άλγεβρα των κατανομών η οποία δημιουργείται από τις επαναλαμβανόμενες αγκύλες Lie. Στα πλαίσια αυτής ελέγχεται η συμπεριφορά των κατανομών, οι οποίες διαχωρίζονται σε ολοκληρώσιμες και bracket generating. Οι τελευταίες παράγουν τον εφαπτόμενο χώρο της πολλαπλότητας και αποτελούν βασική προϋπόθεση για να εφοδιαστεί η πολλαπλότητα με μια υπο-Riemannian δομή. Με αυτή τη δομή ορίζεται η υπο-Riemannian απόσταση από την οποία φτιάχνεται η βάση μιας τοπολογίας που συμπίπτει με τη φυσική τοπολογία της πολλαπλότητας. Σε αυτήν την τοπολογία ορίζονται τα φύλλα του συνόλου προσβασιμότητας. Επιπλέον, δίνεται μια απάντηση και στο πρόβλημα της ελεγξιμότητας, που διαπραγματεύεται η Θεωρία Ελέγχου. Τέλος, γίνεται αναφορά στις γεωδαισιακές εξισώσεις, όπως αυτές ορίζονται στο συνεφαπτόμενο ινώδες των τετραγωνικών μορφών, με χαρακτηριστικό παράδειγμα τις γεωδαισιακές που προκύπτουν από την ομάδα του Heisenberg. / --

Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and Θ an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some ''good" properties of the singular distribution Θ. More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the ''interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of ''quasi-smooth" resolution of families of ideal sheaves.- The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution Θ;- The second result is a global resolution if the the singular distribution Θ has leaf dimension 1;- The third result is a local uniformization if the the singular distribution Θ has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.

[pt] Damos exemplos de variedades suaves abertas que não podem ser folhas de nenhuma folheação riemanniana, nem de qualquer folheação transver- salmente homotética, de variedade compacta. Também apresentamos uma nova classe de não folhas de folheação C0 de codimensão um de variedades compactas, as variedades não periódica em homotopia, usando grupos de homotopia de dimensão maior. Finalmente generalizamos ligeiramente os exemplos de não folhas de Ghys, Inaba et al. e Attie-Hurder, com uma demonstração baseada na recorrência de blocos na variedade. / [en] We give examples of open smooth manifolds that cannot be leaves of any riemannian or transversely homothetic foliation of a compact manifold. We also present a new class of non-leaves of C0 codimension one foliations, manifolds that are non-periodic in homotopy, using higher-dimensional homotopy groups. Finally we give small generalizations of the non-leaf examples of Ghys, Inaba et al., and Attie-Hurder, with a proof based on the recurrence of blocks in the manifold.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Estudamos folheaÃÃes de formas espaciais por hipersuperfÃcies completas, sob certas condiÃÃes sobre as suas curvaturas mÃdias de ordem superior. Em particular, no espaÃo euclidiano obtemos um Teorema tipo-Bernstein para grÃficos cujas curvaturas mÃdia e escalar nÃo mudam de sinal (podendo ser nÃo constantes). NÃs tambÃm estabelecemos a nÃo existÃncia de folheaÃÃes da esfera padrÃo cujas folhas sÃo completas e tÃm curvatura escalar constante, alargando assim um teorema de Barbosa, Kenmotsu e Oshikiri. Para o caso mais geral de folheaÃÃes r-mÃnimas do espaÃo euclidiano, possivelmente com um conjunto singular, somos capazes de invocar um teorema de D. Ferus para dar condiÃÃes sob as quais as folhas nÃo-singulares sÃo folheadas por hiperplanos. / We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of r-minimal foliations of the Euclidean space, possibly with a singular set, we are able to invoke a theorem of Ferus to give conditions under which the nonsigular leaves are foliated by hyperplanes.

A geometria de teias dedica-se ao estudo de invariantes locais para uma determinada configuração de folheações. Uma d-teia é uma coleção de folheações que estão em posição geral. Desta forma, uma d-teia plana, definida em \'R POT.2\' ou \'C POT.2\', nada mais é que uma família de d folheações por curvas. Apresentamos neste trabalho os principais conceitos da teoria clássica de teias, iniciada por W. Blaschke por volta de 1930, bem como uma abordagem atual utilizada no estudo de teias planas. São abordados dois tipos de problemas importantes na teoria: os problemas de linearização e de algebrização de teias. Provamos um resultado clássico no que concerne ao problema de linearização, e um resultado de algebrização de teias empregando métodos desenvolvidos mais recentemente / Web geometry is devoted to the study of local invariants of a certain configuration of foliations. A d-web is a collection of foliations in general position. Therefore, a d-web defined in \'R POT. 2\' or \'C POT. 2\' is just a family of d foliations by curves. We present in this work the main concepts of classical theory of webs, initiated by W. Blaschke around 1930, as well as newer methods used in the study of plane webs. We approach two important types of problems in the theory: problems of linearization and that of algebrization of webs. We prove a classical result concerning the linearization problem, and a result of algebrization of webs using recently developed methods

Given a Geometric Lorenz Flow X on \'R POT. n+2\' of class \'C POT. k+1\'; by definition there exists a Poincaré map \'P IND. X\' of class \'C POT. k+1\'; often so-called Lorenz-type map [ABS83]. The main purpose in this dissertation is to show that under certain conditions the Lorenz-type map \'P IND.X\' can be associate to it a one-dimensional transformation \'f IND. X\' of class \'C POT. k\' (defined on an interval). This association is so-called the reduction transformation R; so we have \'RP IND. X\' = \'f IND. X\'. This association would allow us to study the dynamical properties for the original flow using techniques of one-dimensional dynamics of class \'C POT. k\' / Dado um Fluxo Geométrico de Lorenz X em \'R POT. n+2\' de classe \'C POT. k+1\'; por definição, existe uma aplicação de Poincaré \'P IND. X\'\' de classe \'C POT. k+1\'; frequentemente chamado aplicação do tipo Lorenz [ABS83]. O objetivo principal desta tese é mostrar que, sob certas condições a aplicação do tipo Lorenz \'P IND. X\' pode ser associado a ele uma transformação unidimensional \'f IND. X\' de classe \'C POT. k\' (definida em um intervalo). Esta associação é chamada de transformação de redução R; assim temos que , \'RP IND. X\' = \'f IND. X\'. Esta associação nos permitiria estudar as propriedades dinâmicas do fluxo original utilizando técnicas da dinâmica unidimensional de classe \'C POT. k\'

Orientador: Paulo Régis Caron Ruffino / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T11:21:58Z (GMT). No. of bitstreams: 1 Costa_PauloHenriquePereirada_D.pdf: 1382380 bytes, checksum: 975aac3916932e92b8fe92b185b6eb9f (MD5) Previous issue date: 2013 / Resumo: Seja (M,?) uma variedade Riemanniana compacta folheada. Consideramos uma família de semigrupos Feller compatível em C(Mn) associada as leis de um processo Markoviano de n-pontos. Com algumas condições (Le Jan e Raimond [34]) existe um fluxo estocástico de aplicações mensuráveis em M. Estudamos aqui a degenerescência desses semigrupos tais que o fluxo de aplicações seja folheado, ou seja, cada trajetória permanece na folha em que começou q.s. e portanto cria uma obstrução geométrica natural para a coalescência de trajetórias em folhas distintas. Como uma aplicação dessa teoria, um princípio de médias é provado para uma perturbação de primeira ordem transversal as folhas. Estimativas de taxas de convergências também são dadas / Abstract: Let (M,?) be a compact Riemannian foliated manifold. We consider a family of compatible Feller semigroups in C(Mn) associated to laws of the n-point motion. Under some assumptions (Le Jan and Raimond [34]) there exists a stochastic flow of measurable mappings in M. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated / Doutorado / Matematica / Doutor em Matemática