The action spectrum and C^0 symplectic topology

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.

V2: Changes to the exposition, 28 pages

References in corpus (4)

• Bounds on spectral norms and barcodes
• Barcodes and area-preserving homeomorphisms
• Rigidity of mapping class group actions on $S^1$
• On $C^0$-continuity of the spectral norm for symplectically non-aspherical manifolds

Cited by in corpus (11)

• Bounds on spectral norms and barcodes
• Should $N$-body integrators be symplectic everywhere in phase space?
• Barcodes and area-preserving homeomorphisms
• Proof of the simplicity conjecture
• A Hölder-type inequality for the $C^0$ distance and Anosov-Katok pseudo-rotations
• On $C^0$-continuity of the spectral norm for symplectically non-aspherical manifolds
• Hamiltonian no-torsion
• Topological Persistence in Geometry and Analysis
• Completeness of derived interleaving distances and sheaf quantization of non-smooth objects
• Approximate Identities and Lagrangian Poincaré Recurrence
• The spectral diameter of a Liouville domain