On the growth rate of Leaf-Wise intersections

Abstract

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 $\Lambda M$ is "complicated", if $\Sigma \subseteq T^*M$Σ⊆ T*M is a non-degenerate fibrewise starshaped hypersurface and $\varphi \in \mathrm{Ham}_c (T^*M,\omega)$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $\varphi$ in $\Sigma$ grows exponentially in time. Concrete examples of such manifolds are $(S^2 \times S^2)\#(S^2\#S^2)$, $\mathbb{T}^4\#\mathbb{C}P^2$, or any surface of genus greater than one.