Normalization of the Hamiltonian and the action spectrum

In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold $(M,\omega)$ canonically relates the action spectra of different normalized Hamiltonians on {\it arbitrary} symplectic manifolds $(M,\omega)$. The natural class of normalized Hamiltonians consists of those whose mean value is zero for the closed manifold, and those which are compactly supported in $\text{Int} M$ 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 [Oh4].