Quantitative contact Hamiltonian dynamics

This paper presents a systematic quantitative study of contact rigidity phenomena based on the contact Hamiltonian Floer theory established in [W. J. Merry and I. Uljarević, Maximum principles in symplectic homology, Israel J. Math. 229 (2019) 39–65]. Our quantitative approach applies to arbitrary admissible contact Hamiltonian functions on the contact boundary [Formula: see text] of a weakly[Formula: see text]-monotone symplectic manifold [Formula: see text]. 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: contact big fiber theorem, sufficient conditions for orderability, 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.