  Inspired by Le Calvez' theory of transverse foliations for dynamical systems
of surfaces, we introduce a dynamical invariant, denoted by N, for Hamiltonians
of 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 N on autonomous Hamiltonians thus establishing a certain
uniqueness result for spectral invariants, we obtain a "Max Formula" for
spectral invariants on aspherical manifolds, give a very simple description of
the Entov-Polterovich quasi-state on aspherical surfaces and characterize the
heavy and super-heavy subsets of such surfaces.
