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Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in $\mathbb{R}^4$
 
Naiara V. de Paulo Cidade Universitária, São Paulo, Brazil
Pedro A. S. Salomão Cidade Universitária, São Paulo, Brazil
Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in R^4
eBook ISBN:  978-1-4704-4373-3
Product Code:  MEMO/252/1202.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in R^4
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Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in $\mathbb{R}^4$
Naiara V. de Paulo Cidade Universitária, São Paulo, Brazil
Pedro A. S. Salomão Cidade Universitária, São Paulo, Brazil
eBook ISBN:  978-1-4704-4373-3
Product Code:  MEMO/252/1202.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2522018; 105 pp

    In this article the authors study Hamiltonian flows associated to smooth functions \(H:\mathbb R^4 \to \mathbb R\) restricted to energy levels close to critical levels. They assume the existence of a saddle-center equilibrium point \(p_c\) in the zero energy level \(H^{-1}(0)\). The Hamiltonian function near \(p_c\) is assumed to satisfy Moser's normal form and \(p_c\) is assumed to lie in a strictly convex singular subset \(S_0\) of \(H^{-1}(0)\). Then for all \(E \gt 0\) small, the energy level \(H^{-1}(E)\) contains a subset \(S_E\) near \(S_0\), diffeomorphic to the closed \(3\)-ball, which admits a system of transversal sections \(\mathcal F_E\), called a \(2-3\) foliation. \(\mathcal F_E\) is a singular foliation of \(S_E\) and contains two periodic orbits \(P_2,E\subset \partial S_E\) and \(P_3,E\subset S_E\setminus \partial S_E\) as binding orbits. \(P_2,E\) is the Lyapunoff orbit lying in the center manifold of \(p_c\), has Conley-Zehnder index \(2\) and spans two rigid planes in \(\partial S_E\). \(P_3,E\) has Conley-Zehnder index \(3\) and spans a one parameter family of planes in \(S_E \setminus \partial S_E\). A rigid cylinder connecting \(P_3,E\) to \(P_2,E\) completes \(\mathcal F_E\). All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to \(P_2,E\) in \(S_E\setminus \partial S_E\) follows from this foliation.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Proof of the main statement
    • 3. Proof of Proposition
    • 4. Proof of Proposition
    • 5. Proof of Proposition
    • 6. Proof of Proposition
    • 7. Proof of Proposition -${\rm i})$
    • 8. Proof of Proposition -ii)
    • 9. Proof of Proposition -iii)
    • A. Basics on pseudo-holomorphic curves in symplectizations
    • B. Linking properties
    • C. Uniqueness and intersections of pseudo-holomorphic curves
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2522018; 105 pp

In this article the authors study Hamiltonian flows associated to smooth functions \(H:\mathbb R^4 \to \mathbb R\) restricted to energy levels close to critical levels. They assume the existence of a saddle-center equilibrium point \(p_c\) in the zero energy level \(H^{-1}(0)\). The Hamiltonian function near \(p_c\) is assumed to satisfy Moser's normal form and \(p_c\) is assumed to lie in a strictly convex singular subset \(S_0\) of \(H^{-1}(0)\). Then for all \(E \gt 0\) small, the energy level \(H^{-1}(E)\) contains a subset \(S_E\) near \(S_0\), diffeomorphic to the closed \(3\)-ball, which admits a system of transversal sections \(\mathcal F_E\), called a \(2-3\) foliation. \(\mathcal F_E\) is a singular foliation of \(S_E\) and contains two periodic orbits \(P_2,E\subset \partial S_E\) and \(P_3,E\subset S_E\setminus \partial S_E\) as binding orbits. \(P_2,E\) is the Lyapunoff orbit lying in the center manifold of \(p_c\), has Conley-Zehnder index \(2\) and spans two rigid planes in \(\partial S_E\). \(P_3,E\) has Conley-Zehnder index \(3\) and spans a one parameter family of planes in \(S_E \setminus \partial S_E\). A rigid cylinder connecting \(P_3,E\) to \(P_2,E\) completes \(\mathcal F_E\). All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to \(P_2,E\) in \(S_E\setminus \partial S_E\) follows from this foliation.

  • Chapters
  • 1. Introduction
  • 2. Proof of the main statement
  • 3. Proof of Proposition
  • 4. Proof of Proposition
  • 5. Proof of Proposition
  • 6. Proof of Proposition
  • 7. Proof of Proposition -${\rm i})$
  • 8. Proof of Proposition -ii)
  • 9. Proof of Proposition -iii)
  • A. Basics on pseudo-holomorphic curves in symplectizations
  • B. Linking properties
  • C. Uniqueness and intersections of pseudo-holomorphic curves
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.