PFH spectral invariants and $C^\infty$ closing lemmas

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that any area-preserving diffeomorphism of the torus is $C^\infty$ close to an area-preserving diffeomorphism with dense periodic orbits. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time $\delta$ a periodic orbit must appear of period $O(\delta^{-1})$. We also prove a"Weyl law"describing the asymptotic behavior of PFH spectral invariants.