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Systems of Transversal Sections near Critical Energy Levels of Hamiltonian Systems in $\mathbb R^4$


About this Title

Naiara V. de Paulo and Pedro A. S. Salomão

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 252, Number 1202
ISBNs: 978-1-4704-2801-3 (print); 978-1-4704-4373-3 (online)
DOI: https://doi.org/10.1090/memo/1202
Published electronically: January 25, 2018
Keywords:Hamiltonian dynamics, systems of transversal sections, pseudo-holomorphic curves, periodic orbits, homoclinic orbits

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Proof of the main statement
  • Chapter 3. Proof of Proposition 2.1
  • Chapter 4. Proof of Proposition 2.2
  • Chapter 5. Proof of Proposition 2.8
  • Chapter 6. Proof of Proposition 2.9
  • Chapter 7. Proof of Proposition 2.10-$\rm i)$
  • Chapter 8. Proof of Proposition 2.10-ii)
  • Chapter 9. Proof of Proposition 2.10-iii)
  • Appendix A. Basics on pseudo-holomorphic curves in symplectizations
  • Appendix B. Linking properties
  • Appendix C. Uniqueness and intersections of pseudo-holomorphic curves

Abstract


In this article we study Hamiltonian flows associated to smooth functions restricted to energy levels close to critical levels. We assume the existence of a saddle-center equilibrium point in the zero energy level . The Hamiltonian function near is assumed to satisfy Moser's normal form and is assumed to lie in a strictly convex singular subset of . Then for all small, the energy level contains a subset near , diffeomorphic to the closed -ball, which admits a system of transversal sections , called a foliation. is a singular foliation of and contains two periodic orbits and as binding orbits. is the Lyapunoff orbit lying in the center manifold of , has Conley-Zehnder index and spans two rigid planes in . has Conley-Zehnder index and spans a one parameter family of planes in . A rigid cylinder connecting to completes . All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to in follows from this foliation.

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