Hamiltonian Floer theory on surfaces

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $\Sigma$ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to every collection of capped $1$-periodic orbits which is `maximally unlinked relative the Morse range' a singular foliation on $S^1 \times \Sigma$ which is positively transverse to the vector field $\partial_t \oplus X^H$ and which is assembled in a straight-forward way from the relevant Floer moduli spaces. This provides a Floer-theoretic method for producing foliations of the type which appear in Le Calvez's theory of positively transverse foliations for surface homeomorphisms. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in $CF_*(H,J)$, and which lie in the image of some chain-level PSS map. This leads to the definition of a novel family of spectral invariants which share many of the same formal properties as the Oh-Schwarz spectral invariants, and we compute the novel spectral invariant associated to the fundamental class in entirely dynamical terms. This significantly extends a project initiated by Humili\`{e}re-Le Roux-Seyfaddini in arXiv:1502.03834.Comment: Significant revisions. Main foliation results unaffected, but significant changes to the results concerning the spectral invariants, correcting an error in the previous version. New results and discussions added, and significant changes made to the argument structure to unify presentation of the techniques. 68 pages. Comments welcom