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-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, a partial result on the zero-infinity conjecture, and 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.