Combinatorics of linear stability for Hamiltonian systems in arbitrary dimension

We address the general problem of studying linear stability and bifurcations of periodic orbits for Hamiltonian systems of arbitrary degrees of freedom. We study the topology of the GIT sequence introduced by the first author and Urs frauenfelder, in arbitrary dimension. In particular, we note that the combinatorics encoding the linear stability of periodic orbits is governed by a quotient of the associahedron. Our approach gives a topological/combinatorial proof of the classical Krein--Moser theorem, and refines it for the case of symmetric orbits.