C0-Limits of Hamiltonian Paths and the Oh-Schwarz Spectral Invariants

In this article, we study the behavior of the Oh-Schwarz spectral invariants under C-0-small perturbations of the Hamiltonian flow. We show that if two Hamiltonians G, H vanish on a small ball and if their flows are sufficiently C-0-close, then vertical bar rho(G; a) - rho(H; a)vertical bar <= Cd-C0(path) (phi(t)(G), pi(t)(H)). Using the above result, we prove that if phi is a sufficiently C-0-small Hamiltonian diffeomorphism on a surface of genus g, then parallel to phi parallel to(gamma) <= C(d(C0)(Id, phi))(2-2g-1) hence establishing C-0-continuity of the spectral norm on surfaces. We also present applications of the above results to the theory of Calabi quasimorphisms and improve a result of Entov et al. [9]. In the final section of the paper, we use our results to answer a question of Y.-G. Oh about spectral Hamiltonian homeomorphisms.