Trajectories of the twist map with minimal action and connection with incommensurate structures

Abstract

The theorems which have been obtained for the ground states of the Frenkel-Kontorova model are shown to have applications for invertible area-preserving twist maps of the cylinder onto itself. It allows to predict the existence of quasi-periodic trajectories and of periodic cycles. The topology of these trajectories is described in details. We also show as a consequence of these theorems that when there exists an invariant continuous curve which is not homotopic to zero, its trajectories correspond to absolute minimum of the action.