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 |
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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 252; 2018; 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.
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Table of Contents
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Chapters
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1. Introduction
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2. Proof of the main statement
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3. Proof of Proposition
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4. Proof of Proposition
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5. Proof of Proposition
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6. Proof of Proposition
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7. Proof of Proposition -${\rm i})$
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8. Proof of Proposition -ii)
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9. Proof of Proposition -iii)
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A. Basics on pseudo-holomorphic curves in symplectizations
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B. Linking properties
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C. Uniqueness and intersections of pseudo-holomorphic curves
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Additional Material
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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.
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Chapters
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1. Introduction
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2. Proof of the main statement
-
3. Proof of Proposition
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4. Proof of Proposition
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5. Proof of Proposition
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6. Proof of Proposition
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7. Proof of Proposition -${\rm i})$
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8. Proof of Proposition -ii)
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9. Proof of Proposition -iii)
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A. Basics on pseudo-holomorphic curves in symplectizations
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B. Linking properties
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C. Uniqueness and intersections of pseudo-holomorphic curves