Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Abstract

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces, we introduce a dynamical invariant, denoted by N, for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces. Along the way, we obtain several results of independent interest: We show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with N on autonomous Hamiltonians thus establishing a certain uniqueness result for spectral invariants, we obtain a "Max Formula" for spectral invariants on aspherical manifolds, give a very simple description of the Entov-Polterovich quasi-state on aspherical surfaces and characterize the heavy and super-heavy subsets of such surfaces.

94 pages, 16 figures, 7 footnotes

References in corpus (2)

• Quasi-morphisms and quasi-states in symplectic topology
• Superheavy Lagrangian immersion in 2-torus

Cited by in corpus (10)

• Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds
• Superheavy Lagrangian immersion in 2-torus
• Symplectic structure perturbations and continuity of symplectic invariants
• Fixed point sets of isotopies on surfaces
• A characterization of heaviness in terms of relative symplectic cohomology
• Periodic points of rational area-preserving homeomorphisms
• The spectral diameter of a Liouville domain
• Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces
• Spectral invariants of distance functions
• Hamiltonian Floer theory on surfaces