Finite energy foliations of tight three-spheres and Hamiltonian dynamics

Abstract

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 1 S 2 ≡ RP 3 via its double covering S 3 , and also nondegenerate Hamiltonian systems in R 4 restricted to sphere-like energy surfaces of contact type.

Holomorphic open book decompositions

“…The proof relies on constructing a special finite energy foliation. In special cases they were established by H. Hofer, K. Wysocki and E. Zehnder [25] and by C. Wendl [41], [42]. Proofs usually require a 'starting point', i.e.…”

Paires de structures de contact sur les variétés de dimension trois

“…Remarque 7 Dans le cas général, si est sans contact quadratique avec ˙, on peut définir sur V n deux champs de plans transversaux, qui se comportent le long de à la manière du plan tangent à un feuilletage d'énergie finie près de son lieu singulier [12].…”

“…Son pendant a en effet été démontré par Eliashberg et Hofer [4], via une méthode de remplissage par des disques holomorphes. Par certains aspects, les résultats de cet article peuvent être vus comme une version topologique des feuilletages d'énergie finie de Hofer, Wyzocki et Zehnder [12]. La notion de paire positive est également bien adaptée à celle de surface branchée, connue dans le monde intégrable pour rendre compte des propriétés des laminations.…”

Automatic transversality and orbifolds of punctured holomorphic curves in dimension four

“…The transversality criterion ind.uI c/ > c N .uI c/ is clearly satisfied, and thus u lives in a 1 or 2-dimensional family of embedded curves that never intersect each other. These are precisely the curves that appear in J -holomorphic foliations of W , or in the case where W is a symplectization R M , the finite energy foliations of Hofer, Wysocki and Zehnder [HWZ03]. Isolated curves with ind.uI c/ D 0 can also occur in such foliations (surrounded by families of larger index): we shall show for instance that stable, nicely embedded index 0 curves appear as the underlying somewhere injective curves when families of larger index degenerate to multiple covers.…”

“…These moduli spaces arise quite naturally in a geometric setting: they are the building blocks of J -holomorphic foliations, cf. [HWZ03], [Wen08].…”