Quantitative contact Hamiltonian dynamics

TL;DR

This paper develops a comprehensive contact spectral invariant theory for contact Hamiltonian dynamics, enabling new fundamental results and introducing novel persistence modules called gapped modules, with applications in contact topology.

Contribution

It introduces a systematic quantitative approach to contact Hamiltonian Floer theory, including the development of contact spectral invariants and gapped modules, advancing the understanding of contact rigidity phenomena.

Findings

Established the contact big fiber theorem.

Provided conditions for contact orderability.

Proved existence of translated points.

Abstract

This paper presents a systematic quantitative study of contact rigidity phenomena based on the contact Hamiltonian Floer theory established by Merry-Uljarevi\'c. Our quantitative approach applies to arbitrary admissible contact Hamiltonian functions on the contact boundary $M = \partial W$ of a ${\rm weakly}^{+}$-monotone symplectic manifold $W$. From a theoretical standpoint, we develop a comprehensive contact spectral invariant theory. As applications, the properties of these invariants enable us to establish several fundamental results: the contact big fiber theorem, sufficient conditions for orderability, and the existence results of translated points. Furthermore, we uncover a non-traditional filtration structure on contact Hamiltonian Floer groups, which we formalize through the introduction of a novel type of persistence modules, called gapped modules, that are only parametrized by a partially ordered set. Among the various properties of contact spectral invariants, we highlight that the triangle inequality is derived through an innovative analysis of a pair-of-pants construction in the contact-geometric framework.