Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Abstract

Inspired by Le Calvez’s theory of transverse foliations for dynamical systems on surfaces, we introduce a dynamical invariant, denoted by $\mathcal{N}$, for Hamiltonians on any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces.

Along the way, we obtain several results of independent interest: we show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with $\mathcal{N}$ on autonomous Hamiltonians, thus establishing a certain uniqueness result for spectral invariants; we obtain a “max formula” for spectral invariants on aspherical manifolds; we give a very simple description of the Entov–Polterovich partial quasi-state on aspherical surfaces, and we characterize the heavy and super-heavy subsets of such surfaces.