eBook ISBN:  9781470443733 
Product Code:  MEMO/252/1202.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470443733 
Product Code:  MEMO/252/1202.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

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 saddlecenter 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 \(23\) 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 ConleyZehnder index \(2\) and spans two rigid planes in \(\partial S_E\). \(P_3,E\) has ConleyZehnder 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 pseudoholomorphic curves in symplectizations

B. Linking properties

C. Uniqueness and intersections of pseudoholomorphic curves


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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 saddlecenter 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 \(23\) 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 ConleyZehnder index \(2\) and spans two rigid planes in \(\partial S_E\). \(P_3,E\) has ConleyZehnder 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 pseudoholomorphic curves in symplectizations

B. Linking properties

C. Uniqueness and intersections of pseudoholomorphic curves