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