The Morse index of Chaperon’s generating families

This is an expository paper devoted to the Morse index of Chaperon’s generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott’s iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate the precise dependence of the Morse index from the choice of the generating family associated to a given Hamiltonian diffeomorphism, which will allow to see the Morse index as a Maslov index for the linearized Hamiltonian flow in the symplectic group. We will conclude the survey with a proof that the classical Morse index from Tonelli Lagrangian dynamics coincides with the Maslov index.