Finite energy foliations of tight three-spheres and Hamiltonian dynamics. (English) Zbl 1215.53076

Summary: Surfaces of sections are a classical tool in the study of 3-dimensional dynamical systems. Their use goes back to the work of Poincaré and Birkhoff. In the present paper, we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions. Such a system is established for nondegenerate Reeb flows on the tight 3-sphere by means of pseudoholomorphic curves. The applications cover the nondegenerate geodesic flows on \(T_1S^2\equiv \mathbb RP^3\) via its double covering \(S^3\), and also nondegenerate Hamiltonian systems in \(\mathbb R^4\) restricted to sphere-like energy surfaces of contact type.


53D35 Global theory of symplectic and contact manifolds
37C27 Periodic orbits of vector fields and flows
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
57R30 Foliations in differential topology; geometric theory
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