Secondary invariants of Birkhoff minimizers and heteroclinic orbits

For the generalized Frenkel–Kontorova model on high-dimensional lattices, we study the classification of Birkhoff minimizers with the same rotation vector which is rationally dependent. We introduce secondary invariants to distinguish Birkhoff minimizers with the same rotation vector and show that the collection of Birkhoff minimizers with the same secondary invariants is totally ordered. We also construct heteroclinic connections between neighboring Birkhoff minimizers which are more periodic.