E-raamat: Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in $\mathbb {R}^4$

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.