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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2016.tde-28112016-111054
Document
Author
Full name
Renato Andrielli Laguna
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2016
Supervisor
Committee
Zani, Sergio Luis (President)
Bergamasco, Adalberto Panobianco
Costa, Éder Rítis Aragão
Ebert, Marcelo Rempel
Santos Filho, José Ruidival Soares dos
Title in Portuguese
Hipoeliticidade global para campos vetoriais complexos no plano
Keywords in Portuguese
Campos vetoriais complexos
Hipoeliticidade global
Abstract in Portuguese
Este trabalho consiste em um estudo sobre a propriedade de hipoeliticidade global para campos vetoriais complexos não singulares no plano. As órbitas de Sussmann de um tal campo desempenham um papel fundamental nesta análise. Mostramos que se todas as órbitas são unidimensionais o campo não é globalmente hipoelítico. Quando o campo apresenta uma órbita bidimensional e ao menos uma órbita unidimensional mergulhada também foi demonstrado que este campo não é globalmente hipoelítico. No caso em que o plano é a única órbita, define-se, como em Hounie (1982), uma determinada relação de equivalência entre pontos em que o campo deixa de ser elítico. As classes de equivalência desta relação são homeomorfas a um ponto, a um intervalo compacto ou a uma semirreta. Se todas as classes de equivalência são compactas, o campo é globalmente hipoelítico. Caso haja uma classe de equivalência fechada e homeomorfa a uma semirreta, o campo não é globalmente hipoelítico.
Title in English
Global hypoellipticity for complex vector fields in the plane
Keywords in English
Complex vector fields.
Global hypoellipticity
Abstract in English
This work is a study about global hypoellipticity for nonsingular complex vector fields in the plane. Sussmanns orbits play a fundamental role in this analysis. We show that if all the orbits are one-dimensional then the vector field is not globally hypoelliptic. When there exist a two-dimensional orbit and an embedded one-dimensional one then the vector field is not globally hypoelliptic. In the case when the plane is the only orbit, one defines, as in Hounie (1982), a certain equivalence relation between points where the vector field is not elliptic. The equivalence classes are homeomorphic to a single point, a compact interval or a ray. If all the equivalence classes are compact then the vector field is globally hypoelliptic. If there exists an equivalence class that is closed and homeomorphic to a ray then the vector field is not globally hypoelliptic.
 
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Publishing Date
2016-11-28
 
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