Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

TL;DR

This paper introduces a new dynamical invariant N for Hamiltonians on surfaces, linking it to existing spectral invariants and establishing several key properties and characterizations in symplectic topology.

Contribution

It defines the invariant N for Hamiltonians on surfaces, proves its equivalence to known spectral invariants in specific cases, and establishes uniqueness, max formula, and subset characterizations.

Findings

N coincides with Viterbo and Schwarz spectral invariants on specific surfaces.

A minimal axiomatic spectral invariant must match N on autonomous Hamiltonians.

Established a Max Formula and characterized heavy subsets on aspherical surfaces.

Abstract

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces, 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.