C⁰-limits of Hamiltonian paths and the Oh-Schwarz spectral invariants

2011/09/30 by Sobhan Seyfaddini · 1 citation
Mathematics · #math.SG

paper · pdf · doi:10.1093/imrn/rns191

published as Int. Math. Res. Not. 2013, no. 21, 4920-4960 · 32 pages; v2: minor revisions, to appear in IMRN

arxiv created 2012/08/23 · arxiv updated 2020/01/22

Abstract

In this article we study the behavior of the Oh-Schwarz spectral invariants under C⁰-small perturbations of the Hamiltonian flow. We obtain an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C⁰-distance of its flow from the identity. Using the mentioned estimate we show that, unlike the Hofer norm, the spectral norm is C⁰-continuous on surfaces. We also present applications of the above results to the theory of Calabi quasimorphisms and improve a result of Entov, Polterovich and Py. In the final section of the paper we use our results to answer a question of Y.-G. Oh about spectral Hamiltonian homeomorphisms.

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