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Master's Dissertation
DOI
10.11606/D.45.2011.tde-07072011-150441
Document
Author
Full name
Marcelo Ribeiro de Resende Alves
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2011
Supervisor
Committee
Salomão, Pedro Antonio Santoro (President)
Hryniewicz, Umberto Leone
Ragazzo, Clodoaldo Grotta
Title in Portuguese
Propriedades de dinâmica hamiltoniana em níveis de energia convexos de R4
Keywords in Portuguese
curvas pseudo-holomorfas.
níveis de energia convexos
seções globais
Abstract in Portuguese
A existência de seções globais para uxos é de central importância na teoria de sistemas dinâmicos, pois uma seção global simplica o estudo da dinâmica de um uxo reduzindo-o ao estudo da dinâmica de um difeomorsmo. Apresentamos detalhadamente a construção feita Hofer, Zehnder e Wysocki (em ''The dynamics on a strictly convex energy surface in R4'') de uma seção global para o uxo Hamiltoniano restrito a um nível de energia convexo em R4 . Uma importante consequência da existência dessa seção global é que o uxo Hamiltoniano restrito a um nível de energia convexo em R4 tem 2 ou innitas órbitas periódicas. Essa construção utiliza-se da teoria de curvas pseudo-holomorfas em simplectizações de variedades de contato desenvolvida pelos mesmos autores. Os argumentos apresentados também dão uma nova prova da Conjectura de Weinstein para formas de contato tight em S3 .
Title in English
Properties of the hamiltonian dynamics in convex energy levels of R4
Keywords in English
convex energy levels
global surfaces of section
pseudo-holomorphic curves.
Abstract in English
The existence of global surfaces of section to ows is of central importance in the theory of dynamical systems, as a global surface of section simplies the study of the dynamics of a ow reducing it to the study of the dynamics of a dieomorphism. We present in detail the construction due to Hofer, Wysocki and Zehnder (in ''The dynamics on a strictly convex energy surface in R4'') of a global surface of section for the Hamiltonian ow restricted to a convex energy level in R4 . An important consequence of the existence of the global surface of section is that the Hamiltonian ow restricted to a convex energy level in R4 has either 2 or innitely many periodic orbits. This construction makes use of the theory of pseudo-holomorphic curves in symplectizations of contact manifolds developed by the same authors. The arguments also give a new proof of Weinstein conjecture for tight contact forms in S3 .
 
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Publishing Date
2011-07-20
 
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