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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, thenUsing the above result, we prove that if  ϕ  is a sufficiently  C  0 -small Hamiltonian diffeomorphism on a surface of genus  g , then ∥ ϕ ∥  γ  ≤ C ( d   C  0  ( Id , ϕ )) 2 −2 g −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.

C 0 -Limits of Hamiltonian Paths and the Oh–Schwarz Spectral Invariants