Localization of Floer homology of engulfable topological Hamiltonian loop

Localization of Floer homology is first introduced by Floer \cite{floer:fixed} in the context of Hamiltonian Floer homology. The author employed the notion in the Lagrangian context for the pair $(\phi_H^1(L),L)$ of compact Lagrangian submanifolds in tame symplectic manifolds $(M,\omega)$ in \cite{oh:newton,oh:imrn} for a compact Lagrangian submanifold $L$ and $C^2$-small Hamiltonian $H$. In this article, motivated by the study of topological Hamiltonian dynamics, we extend the localization process for any engulfable Hamiltonian path $\phi_H$ whose time-one map $\phi_H^1$ is sufficiently $C^0$-close to the identity (and also to the case of triangle product), and prove that the value of local Lagrangian spectral invariant is the same as that of global one. Such a Hamiltonian path naturally occurs as an approximating sequence of engulfable topological Hamiltonian loop. We also apply this localization to the graphs $\Graph \phi_H^t$ in $(M\times M, \omega\oplus -\omega)$ and localize the Hamiltonian Floer complex of such a Hamiltonian $H$. We expect that this study will play an important role in the study of homotopy invariance of the spectral invariants of topological Hamiltonian.