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2016 Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces
Vincent Humilière, Frédéric Le Roux, Sobhan Seyfaddini
Geom. Topol. 20(4): 2253-2334 (2016). DOI: 10.2140/gt.2016.20.2253


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.


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Vincent Humilière. Frédéric Le Roux. Sobhan Seyfaddini. "Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces." Geom. Topol. 20 (4) 2253 - 2334, 2016.


Received: 17 March 2015; Revised: 21 August 2015; Accepted: 18 September 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1356.53082
MathSciNet: MR3548467
Digital Object Identifier: 10.2140/gt.2016.20.2253

Primary: 53D40 , 53Dxx
Secondary: 37E30 , 37Exx

Keywords: area-preserving diffeomorphisms , Hamiltonian Floer theory , spectral invariants

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.20 • No. 4 • 2016
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