Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Abstract

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces [25,26], 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.Formal Spectral Invariants: Although the following definition makes sense on any symplectic manifold we will restrict our attention here to the case of a surface Σ which is either the plane R 2 or is closed and aspherical. Definition 3. A function c : C ∞ ([0, 1] × Σ) → R is a formal spectral invariant if it satisfies the following four axioms: 1. (Spectrality) c(H) ∈ spec(H) for all H ∈ C ∞ ([0, 1]×Σ), where spec(H), the spectrum of H, is the set of critical values of the Hamiltonian action, that is, the set of actions of fixed points of φ 1 H .

Spectral invariants for monotone Lagrangians

“…They have been used in numerous deep applications, such as metrics on infinite-dimensional groups of symmetries [Vit92, Sch00, Oh05b, Kha09, Lec08, San10, MZ11, CS12, Zap13a, Zap13b, Sey14]; the symplectic camel problem [Vit92,Thé99]; quasi-morphisms on the Hamiltonian group [EP03,Ost06,Ush11,FOOO11]; quasi-states and symplectic and contact rigidity [EP06, EP09, MVZ12, Zap13a]; orderability and contact nonsqueezing [San11,AM13]; C 0 -symplectic topology [Sey13a,Sey13b,HLS14,HLS15a,HLS15b]; function theory on symplectic manifolds [EPZ07,BEP12], [PR14] and the references therein; quantum measurements and noise [Pol12,Pol14]; surface dynamics [HLRS15]; and contact dynamics [Zap13a]. There are other applications, and it is not feasible to list all of them here, but the above sample should give the reader a feeling of the power of this wonderful tool of symplectic topology.…”

A brief survey of the spectral numbers in floer homology

“…depends only on . It turns out that these two constructions give the same result (see [56,57] for more details; see also [42] for similar relations in a different context and for the surfaces).…”

Fixed point sets of isotopies on surfaces

“…Le Calvez's theorem is a extremely powerful tool to study surface dynamics (see for example the introduction of [LCT15] and the references within). The general existence of maximally unlinked sets is a crucial auxiliary tool for Le Calvez's theorem: it extends its range of action from some specific cases, when the existence of maximally unlinked sets is obvious (for instance when the fixed point set is assumed to be finite, as in Le Calvez's proof of Arnol'd's conjecture in [LC05]) or easy (for instance for diffeomorphisms, see the appendix of [HLRS15]), to the case of every homeomorphism which is isotopic to the identity.…”