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 C⁰-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 C⁰-close, then $|\rho (G;a)-\rho (H;a)|\leq C\ d^{\rm path}_{C^0}\ (\phi^t_G,\phi^t_H)$ Using the above result, we prove that if ϕ is a sufficiently C⁰-small Hamiltonian diffeomorphism on a surface of genus g, then ∥ϕ∥γ≤C(dC⁰(Id,ϕ))^2−2g−1 hence establishing C⁰-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.