2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71057In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian diffeomorphisms on arbitrary, especially on {\it non-exact and non-rational}, compact symplectic manifold $(M,ω)$. To each given time dependent Hamiltonian function $H$ and quantum cohomology class $ 0 \neq a \in QH^*(M)$, we associate an invariant $ρ(H;a)$ which varies continuously over $H$ in the $C^0$-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is `dual' to the given quantum cohomology class $a$ on the covering space $\widetilde Ω_0(M)$ of the contractible loop space $Ω_0(M)$. We call them the {\it Novikov Floer cycles}. We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper.43 pages, In this version, we fill a gap in the proof of spectrality axiom in the previous version and provide a complete proof of the spectraity axiom for the rational symplectic manifolds. A separate paper (math.SG/0406449) deals with the spectrality axiom for the irrational cases. To appear in the volume in honor of Alan Weinstein's 60th BirthdaySymplectic Geometry53D35, 53D40text