On the growth rate of leaf-wise intersections

TL;DR

This paper introduces a new variant of Rabinowitz Floer homology to study the exponential growth rate of leaf-wise intersections on certain complex manifolds, revealing new dynamical properties.

Contribution

It develops a novel Floer homology framework and demonstrates exponential growth of leaf-wise intersections for specific high-genus and connected sum manifolds.

Findings

Exponential growth of leaf-wise intersections on certain manifolds.

New Rabinowitz Floer homology variant suited for growth rate analysis.

Applicable to manifolds with complicated loop spaces.

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 $T^*M$ 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.