Ground states and critical points for generalized Frenkel-Kontorova models in Zd

We consider a multidimensional model of the Frenkel-Kontorova type but we allow non-nearest-neighbour interactions, which satisfy some weak version of ferromagnetism. For every possible frequency vector, we show that there are quasiperiodic ground states which enjoy further geometric properties. The ground states we produce are either bigger or smaller than their integer translates. They are at a bounded distance from the plane wave with the given frequency. The comparison property above implies that the ground states and the translations are organized into laminations. If these leave a gap, we show that there are critical points inside the gap which also satisfy the comparison properties. In particular, given any frequency, we show that either there is a continuous parameter of ground states or there is a ground state and another critical point which is not a ground state. This is a higher dimensional analogue of the criterion of the non-existence of invariant circles if and only if there is a positive Peierls-Nabarro barrier. All the above results are higher dimensional extensions of similar results in Aubry-Mather theory.