Quantitative contact Hamiltonian dynamics

Abstract

This paper presents a systematic quantitative study of contact rigidity phenomena based on the contact Hamiltonian Floer theory established by Merry-Uljarević. Our quantitative approach applies to arbitrary admissible contact Hamiltonian functions on the contact boundary $M=\partial W$ of a ${\mathrm{weakly}}^{+}$-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.

this paper incorporates our earlier preprints arXiv:2309.00527 and arXiv:2503.18750; updating the previous version: the orignianl Theorem 1.7 on the zero-or-infinity conjecture has been removed, the direction of the triangle inequality for the contact spectral invariant has been reversed due to the sign convention, and a new Section 8 on a Poisson bracket inequality has been added