Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Humilière, Vincent¹ ; Le Roux, Frédéric¹ ; Seyfaddini, Sobhan²

¹ Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France
² Département de Mathématiques et Applications de l’École Normale Supérieure, 45 rue d’Ulm, 75320 Paris, France, Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, United States

Geometry & topology, Tome 20 (2016) no. 4, pp. 2253-2334.

Résumé

Inspired by Le Calvez’s theory of transverse foliations for dynamical systems on surfaces, we introduce a dynamical invariant, denoted by $\mathcal{N}$, for Hamiltonians on 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 $\mathcal{N}$ on autonomous Hamiltonians, thus establishing a certain uniqueness result for spectral invariants; we obtain a “max formula” for spectral invariants on aspherical manifolds; we give a very simple description of the Entov–Polterovich partial quasi-state on aspherical surfaces, and we characterize the heavy and super-heavy subsets of such surfaces.

Détail

DOI : 10.2140/gt.2016.20.2253

Classification : 53D40, 53DXX, 37EXX, 37E30
Keywords: spectral invariants, Hamiltonian Floer theory, area-preserving diffeomorphisms

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