Homological Resonances for Hamiltonian Diffeomorphisms and Reeb Flows

We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural requirements. In the case of Reeb flows, this leads to simple expressions (purely in terms of the mean indices) for the mean Euler characteristics. These are invariants of the underlying contact structure which are capable of distinguishing some contact structures that are homotopic but not diffeomorphic.