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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   $\varphi\big|_N$   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.

Leafwise Fixed Points for $C^0$ -Small Hamiltonian Flows