Shelukhin's Hofer distance and a symplectic cohomology barcode for contactomorphisms

TL;DR

This paper introduces a new symplectic cohomology barcode for contactomorphisms, linking it to Shelukhin's Hofer distance, and uses it to derive existence results and spectral invariants for contactomorphisms.

Contribution

It constructs a persistence module for Floer cohomology associated to contactomorphisms and establishes bounds and spectral properties related to Shelukhin's Hofer distance.

Findings

Barcode supported on lengths of translated points

Bound on barcode distance by Shelukhin's Hofer distance

Spectral invariants monotone under positive paths

Abstract

This paper constructs a persistence module of Floer cohomology groups associated to a contactomorphism of the ideal boundary of a Liouville manifold. The barcode (or, bottleneck) distance between the persistence modules is bounded from above by Shelukhin's Hofer distance. Moreover, the barcode is supported (i.e., has spectrum) on the lengths of translated points of the contactomorphism. We use this structure to prove various existence results for translated points and to construct spectral invariants for contactomorphisms which are monotone with respect to positive paths and continuous with respect to Shelukhin's Hofer distance. While this paper was nearing completion, the author was made aware of similar upcoming work by Djordjevi\'c, Uljarevi\'c, Zhang.