Coisotropic Submanifolds and Leaf-wise Fixed Points of Hamiltonian Diffeomorphisms

Let $(M,\omega )$ be a symplectic manifold and let $\varphi :M\to M$ be a Hamiltonian diffeomorphism. During the last twenty years, symplectic topologists intensively studied the following two questions:

1. How many fixed points does $\varphi$ at least have?
2. Given a Lagrangian submanifold $L$ in $M$, how many intersection points do $L$ and $\varphi (L)$ at least have?

Consider now a coisotropic submanifold $Q$ in $M$. A point $x$ in $Q$ is called a leaf-wise fixed point of $\varphi$ if $x$ and $\varphi (x)$ lie in the same isotropic leaf of $Q$. Generalizing the above questions, one may ask if there is a lower bound on the number of leaf-wise fixed points of $\varphi$. The main result of this talk is that under suitable assumptions on $M,\omega ,Q$ and $\varphi$ such a bound is given by the sum of the ${\text{Z}}_2$-Betti numbers of $Q$.