Leaf-wise intersections in coisotropic submanifolds (Q1951665)

The submanifolds of a symplectic manifold whose tangent space contains its symplectic orthogonal are called coisotropic. These symplectic orthogonal subspaces define an integrable distribution tangent to the so-called characteristic foliation. Given a symplectic diffeomorphism and a coisotropic submanifold, the paper under review studies the existence of leaf-wise intersections, i.e., intersection points between a leaf of the characteristic foliation and its image by the symplectic diffeomorphism. In the special case where the coisotropic submanifold is the symplectic manifold itself, leafwise intersections are the fixed points of the diffeomorphism. If the coisotropic submanifold is a connected Lagrangian submanifold, leafwise intersections are intersection points between the Lagrangian and its image. The present paper shows that for every symplectic diffeomorphism \(C^1\)-closed to the identity, there exists a closed 1-form on the coisotropic submanifold whose zeros correspond to the leafwise intersections of the diffeomorphism. It allows to recover and improve slightly a theorem of \textit{J. Moser} [Acta Math. 141, 17--34 (1978; Zbl 0382.53035)] on the number of leafwise intersections.