**Memoirs of the American Mathematical Society**

2018;
105 pp;
Softcover

Print ISBN: 978-1-4704-2801-3

Product Code: MEMO/252/1202

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4373-3
Product Code: MEMO/252/1202.E**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

# Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in \(\mathbb{R}^{4}\)

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*Naiara V. de Paulo; Pedro A. S. Salomão*

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

# Table of Contents

## Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in $\mathbb{R}^{4}$

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Proof of the main statement 1522
- Chapter 3. Proof of Proposition 2.1 2330
- Chapter 4. Proof of Proposition 2.2 3340
- Chapter 5. Proof of Proposition 2.8 4552
- Chapter 6. Proof of Proposition 2.9 4956
- Chapter 7. Proof of Proposition 2.10-𝑖) 5360
- Chapter 8. Proof of Proposition 2.10-ii) 5966
- Chapter 9. Proof of Proposition 2.10-iii) 7178
- Appendix A. Basics on pseudo-holomorphic curves in symplectizations 8794
- Appendix B. Linking properties 93100
- Appendix C. Uniqueness and intersections of pseudo-holomorphic curves 97104
- References 103110
- Back Cover Back Cover1118