The action spectrum and C^0 symplectic topology

TL;DR

This paper proves the C^0 continuity of the spectral norm on Hamiltonian diffeomorphisms for symplectically aspherical manifolds, extends barcodes to Hamiltonian homeomorphisms, and discusses implications for the Arnold conjecture in the C^0 setting.

Contribution

It establishes C^0 continuity of spectral invariants and barcodes for Hamiltonian homeomorphisms on aspherical manifolds, and reformulates the Arnold conjecture accordingly.

Findings

Spectral norm is C^0 continuous on symplectically aspherical manifolds.

Barcodes are extended to Hamiltonian homeomorphisms.

A reformulated Arnold conjecture holds for Hamiltonian homeomorphisms.

Abstract

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.