  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.
