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 ; Kris Wysocki ; Eduard Zehnder 1
@article{10_4007_annals_2003_157_125,
author = {Helmut Hofer and Kris Wysocki and Eduard Zehnder},
title = {Finite energy foliations of tight three-spheres and {Hamiltonian} dynamics},
journal = {Annals of mathematics},
pages = {125--255},
year = {2003},
volume = {157},
number = {1},
doi = {10.4007/annals.2003.157.125},
mrnumber = {1954266},
zbl = {1215.53076},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.125/}
}
TY - JOUR AU - Helmut Hofer AU - Kris Wysocki AU - Eduard Zehnder TI - Finite energy foliations of tight three-spheres and Hamiltonian dynamics JO - Annals of mathematics PY - 2003 SP - 125 EP - 255 VL - 157 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.125/ DO - 10.4007/annals.2003.157.125 LA - en ID - 10_4007_annals_2003_157_125 ER -
%0 Journal Article %A Helmut Hofer %A Kris Wysocki %A Eduard Zehnder %T Finite energy foliations of tight three-spheres and Hamiltonian dynamics %J Annals of mathematics %D 2003 %P 125-255 %V 157 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.125/ %R 10.4007/annals.2003.157.125 %G en %F 10_4007_annals_2003_157_125
Helmut Hofer; Kris Wysocki; Eduard Zehnder. Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Annals of mathematics, Tome 157 (2003) no. 1, pp. 125-255. doi: 10.4007/annals.2003.157.125
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