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 is "complicated", if $\Sigma$ is a non-degenerate fibrewise starshaped hypersurface in $T^{\ast }M$ and $\varphi$ 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 $M$ are the connected sum of two copies of $S^2\times S^2$, the connected sum of $T^4$ and $C{P}^2$, or any surface of genus greater than one.

42 pages