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 $ΛM$ is “complicated”, if $Σ ⊆ T∗M$ is a non-degenerate fibrewise starshaped hypersurface and $ϕ ∈ Ham_c(T∗M,ω)$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $ϕ$ in $Σ$ grows exponentially in time. Concrete examples of such manifolds are $(S^2×S^2)#(S^2#S^2), T^4#CP^2$, or any surface of genus greater than one.

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Published 28 September 2012