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Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Vincent Humilière, Frédéric Le Roux and Sobhan Seyfaddini

Geometry & Topology 20 (2016) 2253–2334

Inspired by Le Calvez’s theory of transverse foliations for dynamical systems on surfaces, we introduce a dynamical invariant, denoted by 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 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.

spectral invariants, Hamiltonian Floer theory, area-preserving diffeomorphisms
Mathematical Subject Classification 2010
Primary: 53D40, 53DXX
Secondary: 37EXX, 37E30
Received: 17 March 2015
Revised: 21 August 2015
Accepted: 18 September 2015
Published: 15 September 2016
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Yasha Eliashberg
Vincent Humilière
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie
4 Place Jussieu
75005 Paris
Frédéric Le Roux
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie
4 Place Jussieu
75005 Paris
Sobhan Seyfaddini
Département de Mathématiques et Applications de l’École Normale Supérieure
45 rue d’Ulm
75320 Paris
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States