Leafwise fixed points for $C^0$-small Hamiltonian flows

Consider a closed coisotropic submanifold $N$ of a symplectic manifold $(M,\omega )$ and a Hamiltonian diffeomorphism $\varphi$ on $M$. The main result of this article states that $\varphi$ has at least the cup-length of $N$ many leafwise fixed points w.r.t. $N$, provided that it is the time-1-map of a global Hamiltonian flow whose restriction to $N$ stays $C^0$-close to the inclusion $N\to M$. If $(\varphi ,N)$ is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of $N$. The nondegeneracy condition is generically satisfied. This appears to be the first leafwise fixed point result in which neither $\varphi {\mid }_N$ is assumed to be $C^1$-close to the inclusion $N\to M$, nor $N$ to be of contact type or regular (i.e., “fibering”). It is optimal in the sense that the $C^0$-condition on $\varphi$ cannot be replaced by the assumption that $\varphi$ is Hofer-small.