Local Homology and Hilbert Subspaces

Abstract

In a Lagrangian system associated to a Lagrangian function L : R/ZxTM → R, a periodic orbit γ : R/Z → M produces a critical point of the action functional in any period n ∈ N. Namely, the iterated curve γ[n] is a critical point of A [n]. As we already saw in Section 4.6, critical point theory allows us to investigate the multiplicity of periodic orbits with prescribed period n. On the other hand, if one is interested in the multiplicity of periodic orbits with any period, some additional arguments are needed in order to recognize when an n-periodic orbit is the iteration of a periodic orbit of smaller period.