On the growth rate of leaf-wise intersections

We define a new variant of Rabinowitz Floer homology that is particularly well suited to studying the growth rate of leaf-wise intersections. We prove that for closed manifolds $M$ whose loop space is "complicated", if $\Sigma$ is a non-degenerate fibrewise starshaped hypersurface in $T^*M$ and $\phi$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $\phi$ in $\Sigma$ grows exponentially in time. Concrete examples of such manifolds $M$ are the connected sum of two copies of $S^2 \times S^2$, the connected sum of $T^4$ and $CP^2$, or any surface of genus greater than one.