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Doctoral Thesis
DOI
10.11606/T.45.2018.tde-04042018-171903
Document
Author
Full name
Alexsandro Schneider
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Salomão, Pedro Antonio Santoro (President)
Oliveira, Rafael Ribeiro Dias Vilela de
Paulo, Naiara Vergian de
Silva, André Vanderlinde da
Zanata, Salvador Addas
Title in Portuguese
Seções globais para fluxos de Reeb dinamicamente convexos em $L(p, 1)$ e folheação $3-2^3$ no Hamiltoniano de Hénon-Heiles
Keywords in Portuguese
Espaços lenticulares
Fluxos de Reeb
Hamiltoniano de Hénon-Heiles
Seções globais
Sistemas globais de seções transversais
Abstract in Portuguese
Neste trabalho, mostramos que fluxos de Reeb dinamicamente convexos em um espaço lenticular $L(p, 1)$, $p>1$, admite uma órbita periódica de Reeb especial $P$ que é o binding de uma decomposição em livro aberto racional, com páginas tipo-disco tal que cada página é uma seção global. O índice de Conley-Zehnder da $p$-ésima iterada de $P$ é $3$. Como corolário, o fluxo de Reeb possui duas ou infinitas órbitas periódicas. Este resultado aplica-se ao Hamiltoniano de Hénon-Heiles, cujo fluxo restrito a energia baixa possui $Z_3$-simetria e define um fluxo de Reeb em $L(3, 1)$. Devido a $Z_4$-simetria aplicamos nosso resultado ao problema lunar de Hill regularizado. Na segunda parte deste trabalho investigamos a existência de uma folheação $3-2^3$ em níveis de energia no sistema Hamiltoniano de Hénon-Heiles, para energia logo acima da crítica. Provamos que certa região de interesse é uma hipersuperfície de contato. Provamos também que o fluxo de Reeb possui uma órbita periódica $Z_3$ simétrica, cujo índice de Conley-Zehnder é $3$ e possui número de auto-enlaçamento $-1$.
Title in English
Global surfaces of section for dynamically convex Reeb flows on $L(p, 1)$ and $3-2^3$ foliation in the Hénon-Heiles Hamiltonian
Keywords in English
Hénon-Heiles Hamiltonian
Lens spaces
Reeb flows
Surfaces of section
Systems of transversal sections
Abstract in English
We show that a dynamically convex Reeb flow on a lens space $L(p, 1)$, $p>1$ admits a special closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages so that each page is a global surface of section. The Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. As a corollary, the Reeb flow has $2$ or infinitely many closed Reeb orbits. This result applies to the Hénon-Heiles Hamiltonian whose flow restricted to low energy levels has $Z_3$-symmetry and descends to $L(3,1)$. Due to a $Z_4$-symmetry we also apply our results to Hill's lunar problem. In the second part of this work we investigate the existence of a $3-2^3$ foliation on energy levels of the Hénon-Heiles Hamiltonian, for energies above the critical one. We show that some region is of contact-type and the Reeb flow has a $Z_3$-symmetric periodic orbit, whose Conley-Zehnder is $3$ and has self-linking number $-1$.
 
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Publishing Date
2018-11-23
 
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