Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez’ theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by N, for Hamiltonians on surfaces (except the sphere). We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the classical spectral invariant. This is joint work with Vincent Humilière and Frédéric Le Roux.