C^0-limits of Hamiltonian paths and the Oh-Schwarz spectral invariants

TL;DR

This paper investigates how Oh-Schwarz spectral invariants behave under small C^0 perturbations of Hamiltonian flows, establishing continuity properties and applications to spectral Hamiltonian homeomorphisms and Calabi quasimorphisms.

Contribution

It introduces a C^0-continuity estimate for spectral invariants on surfaces and applies this to improve existing results and address open questions in Hamiltonian dynamics.

Findings

Spectral invariants are C^0-continuous on surfaces under certain conditions.

The spectral norm differs from the Hofer norm in C^0 topology.

Applications include results on Calabi quasimorphisms and spectral Hamiltonian homeomorphisms.

Abstract

In this article we study the behavior of the Oh-Schwarz spectral invariants under C^0-small perturbations of the Hamiltonian flow. We obtain an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C^0-distance of its flow from the identity. Using the mentioned estimate we show that, unlike the Hofer norm, the spectral norm is C^0-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.