Elliptic bindings for dynamically convex Reeb flows on the real projective three-space

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• 作者：Umberto L. Hryniewicz ; Pedro A. S. Salomão
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：55
• 期：2
• 全文大小：1,029 KB
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• 作者单位：Umberto L. Hryniewicz (1)
  Pedro A. S. Salomão (2)
  
  1. Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Rio de Janeiro, RJ, 21941-909, Brazil
  2. Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP, 05508-090, Brazil
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

The first result of this paper is that every contact form on \(\mathbb {R}P^3\) sufficiently \(C^\infty \)-close to a dynamically convex contact form admits an elliptic–parabolic closed Reeb orbit which is 2-unknotted, has self-linking number \(-1/2\) and transverse rotation number in (1 / 2, 1]. Our second result implies that any p-unknotted periodic orbit with self-linking number \(-1/p\) of a dynamically convex Reeb flow on a lens space of order p is the binding of a rational open book decomposition, whose pages are global surfaces of section. As an application we show that in the planar circular restricted three-body problem for energies below the first Lagrange value and large mass ratio, there is a special link consisting of two periodic trajectories for the massless satellite near the smaller primary—lunar problem—with the same contact-topological and dynamical properties of the orbits found by Conley (Commun Pure Appl Math 16:449–467, 1963) for large negative energies. Both periodic trajectories bind rational open book decompositions with disk-like pages which are global surfaces of section. In particular, one of the components is an elliptic–parabolic periodic orbit.