Journal of Symplectic Geometry

Volume 10 (2012)

Number 4

On the growth rate of leaf-wise intersections

Pages: 601 – 653

DOI: http://dx.doi.org/10.4310/JSG.2012.v10.n4.a5

Authors

Leonardo Macarini (Universidade Federal do Rio de Janeiro, Instituto de Matemática, Cidade Universitária, Rio de Janeiro, Brazil)

Will J. Merry (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)

Gabriel P. Paternain (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)

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 $ΛM$ 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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