Normalization of the Hamiltonian and the action spectrum

Abstract

In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold (M, w) canonically relate the action spectra of different. normalized Hamiltonians on arbitrary symplectic manifolds (M, w). The natural classes of normalized Hamiltonians consist of those whose mean value is zero for the closed manifold, and those, which are compactly supported in IntM for the open manifold, We also study the effect of the action spectrum under the pi(1) of Hamiltonian diffeomorphism group. This forms a foundational basis for our study of spectral invariants of the Hamiltonian diffeomorphism in [8].