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OverviewIn 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. Full Product DetailsAuthor: Naiara V. de Paulo , Pedro A.S. SalomaoPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.220kg ISBN: 9781470428013ISBN 10: 1470428016 Pages: 105 Publication Date: 30 April 2018 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Temporarily unavailable  The supplier advises that this item is temporarily unavailable. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out to you. Table of ContentsIntroduction Proof of the main statement Proof of Proposition 2.1 Proof of Proposition 2.2 Proof of Proposition 2.8 Proof of Proposition 2.9 Proof of Proposition 2.10-i) Proof of Proposition 2.10-ii) 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 ReferencesReviewsAuthor InformationNaiara V. de Paulo, Cidade Universitaria, Sao Paulo, Brazil. Pedro A. S. Salomao, Cidade Universitaria, Sao Paulo, Brazil. Tab Content 6Author Website:Countries AvailableAll regions |
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