On Cap-Decorated Floer Persistence modules

Abstract: For a closed symplectically aspherical manifold, we introduce cap-decorated Floer persistence by refining the action-filtered Hamiltonian Floer persistence module with layers defined by iterated actions of homogeneous cohomology ideals. We prove functoriality and stability of these layers, and construct a universal cap-profile pseudodistance bounded above by Hofer distance. As an application, we obtain new invariants of Hamiltonian conjugacy classes, detecting symplectic mapping-class displacement phenomena invisible to ordinary Floer barcodes and spectral invariants. In particular, for a genus-two surface we construct Hamiltonian diffeomorphisms with identical ordinary Floer data but positive separation in the Hamiltonian-conjugacy quotient.