  I study classes of generalized Frenkel-Kontorova models whose potentials are
given by almost-periodic functions which are closely related to aperiodic
Delone sets of finite local complexity. Since such Delone sets serve as good
models for quasicrytals, this setup presents Frenkel-Kontorova models for the
type of aperiodic crystals which have been discovered since Shechtman's
discovery of quasicrystals. Here I consider models with configurations
$u:\mathbb{Z}^r \rightarrow \mathbb{R}^d$, where $d$ is the dimension of the
quasicrystal, for any $r$ and $d$. The almost-periodic functions used for
potentials are called pattern-equivariant and I show that if the interactions
of the model satisfies a mild $C^2$ requirement, and if the potential satisfies
a mild non-degeneracy assumption, then there exist equilibrium configurations
of any prescribed rotation rotation number/vector/plane. The assumptions are
general enough to satisfy the classical Frenkel-Kontorova models and its
multidimensional analogues. The proof uses the idea of the anti-integrable
limit.
