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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.

MSC:

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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