#### Volume 20, issue 4 (2016)

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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
##### Abstract

Inspired by Le Calvez’s theory of transverse foliations for dynamical systems on surfaces, we introduce a dynamical invariant, denoted by $\mathsc{N}$, 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 $\mathsc{N}$ 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.

##### Keywords
spectral invariants, Hamiltonian Floer theory, area-preserving diffeomorphisms
##### Mathematical Subject Classification 2010
Primary: 53D40, 53DXX
Secondary: 37EXX, 37E30
##### Publication
Received: 17 March 2015
Revised: 21 August 2015
Accepted: 18 September 2015
Published: 15 September 2016
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Yasha Eliashberg
##### Authors
 Vincent Humilière Institut de Mathématiques de Jussieu Université Pierre et Marie Curie 4 Place Jussieu 75005 Paris France Frédéric Le Roux Institut de Mathématiques de Jussieu Université Pierre et Marie Curie 4 Place Jussieu 75005 Paris France Sobhan Seyfaddini 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