Quantitative characterization in contact Hamiltonian dynamics -- I

TL;DR

This paper develops a framework for defining and analyzing contact spectral invariants using Floer theory, establishing stability and triangle inequalities without conformal factors, advancing quantitative contact geometry.

Contribution

It introduces a gapped persistence module for contact Hamiltonian systems, enabling new numerical Floer invariants and properties like stability and triangle inequalities.

Findings

Proves stability of contact spectral invariants under the Shelukhin-Hofer norm.

Establishes a triangle inequality for contact spectral invariants.

Develops a novel analysis on pair-of-pants in contact Floer homology.

Abstract

Based on the contact Hamiltonian Floer theory established by Will J. Merry and the second author that applies to any admissible contact Hamiltonian system $(M, \xi = \ker \alpha, h)$, where $h$ is a contact Hamiltonian function on a Liouville fillable contact manifold $(M, \xi = \ker \alpha)$, we associate a persistence module to $(M, \xi, h)$, called a gapped module, that is parametrized only by a partially ordered set. It enables us to define various numerical Floer-theoretic invariants. In this paper, we focus on the contact spectral invariants and their applications. Several key properties are proved, which include stability with respect to the Shelukhin-Hofer norm in contact geometry and a triangle inequality of contact spectral invariants. In particular, our stability property does not involve any conformal factors; our triangle inequality is derived from a novel analysis on pair-of-pants in the contact Hamiltonian Floer homology. While this paper was nearing completion, the authors were made aware of upcoming work by Dylan Cant, where a similar persistence module for contact Hamiltonian dynamics was constructed.