Finite energy foliations of tight three-spheres and Hamiltonian dynamics


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 $T_1S^2\equiv \mathbb{R}P^3$ via its double covering $S^3$, and also nondegenerate Hamiltonian systems in $\mathbb{R}^4$ restricted to sphere-like energy surfaces of contact type.


Helmut Hofer

Current address:

School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 Kris Wysocki

Current address:

Mathematics Department, Penn State University, University Park, State College, PA 16802 Eduard Zehnder

Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland