Selective Floer cohomology for contact vector fields

TL;DR

This paper introduces a new invariant called selective Floer cohomology for contact vector fields, which captures their dynamics and leads to results on positive orbits and symplectic non-squeezing.

Contribution

It develops a novel persistence module invariant for contact vector fields on Liouville boundaries, connecting contact dynamics with symplectic topology.

Findings

Proves existence of positive orbits for certain contact vector fields.

Recovers the non-squeezing theorem of Eliashberg, Kim, and Polterovich.

Abstract

This paper associates a persistence module to a contact vector field $X$ on the ideal boundary of a Liouville manifold. The persistence module measures the dynamics of $X$ on the region $\Omega$ where $X$ is positively transverse to the contact distribution. The colimit of the persistence module depends only on the domain $\Omega$ and is a variant of the selective symplectic homology introduced by the second named author. As an application we prove existence of positive orbits for certain classes of contact vector fields. Another application of this invariant is that we recover the famous non-squeezing result of Eliashberg, Kim, and Polterovich.