TAILIEUCHUNG - Đề tài " 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 Poincar´ and Birkhoff. e 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. | Annals of Mathematics Finite energy foliations of tight three-spheres and Hamiltonian dynamics By H. Hofer K. Wysocki and E. Zehnder Annals of Mathematics 157 2003 125 257 Finite energy foliations of tight three-spheres and Hamiltonian dynamics By H. Hofer K. Wysocki and E. Zehnder Abstract Surfaces of sections are a classical tool in the study of 3-dimensional dynamical systems. Their use goes back to the work of Poincare 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 T1S2 RP3 via its double covering S3 and also nondegenerate Hamiltonian systems in R4 restricted to sphere-like energy surfaces of contact type. Contents 1. Introduction . Concepts from contact geometry and Reeb flows . Finite energy spheres in S3 . Finite energy foliations . Stable finite energy foliations the main result . Outline of the proof . Application to dynamical systems 2. The main construction . The problem M . Gluing almost complex half cylinders over contract boundaries . Embeddings into cP2 the problems V and W . Pseudoholomorphic spheres in cP2 The research of the first author was partially supported by an NSF grant a Clay scholarship and the Wolfensohn Foundation. The research of the second author was partially supported by an Australian Research Council grant. The research of the third author was partially supported by TH-project. 126 H. HOFER K. WYSOCKI AND E. ZEHNDER 3. Stretching the neck 4. The bubbling off tree 5. Properties of bubbling off trees . Fredholm indices . Analysis of bubbling off trees 6. Construction of a stable finite energy foliation . Construction of a dense set of leaves . Bubbling off as mk m .

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