Symplectic homology of displaceable Liouville domains and Leafwise intersection points

TL;DR

This paper proves that the symplectic homology of displaceable Liouville domains vanishes, but filtered symplectic homologies can still produce leafwise intersection points with period-dependent energy, revealing nuanced dynamical properties.

Contribution

It demonstrates the vanishing of symplectic homology for displaceable Liouville domains and introduces a method to find leafwise intersection points with period-dependent energy.

Findings

Symplectic homology of displaceable Liouville domains vanishes.

Filtered symplectic homologies can detect leafwise intersection points.

Leafwise intersection points have energy varying with period.

Abstract

In this note we prove that the symplectic homology of a Liouville domain W displaceable in the symplectic completion vanishes. Nevertheless if the Euler characteristic of (W,\p W) is odd, the filtered symplectic homologies of W do not vanish and give rise to leafwise intersection points on the symplectic completion of W for a perturbation displacing $W$ from itself. In contrast to the existing results we can find a leafwise intersection point for a given period but its energy varies by period instead.