  Monotone variational recurrence relations such as the Frenkel-Kontorova
lattice, arise in solid state physics, conservative lattice dynamics and as
Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory
guarantees the existence of solutions of every rotation number. They are the
action minimizers that constitute the Aubry-Mather set. When the rotation
number is irrational, the Aubry-Mather set is either connected or a Cantor set.
A connected Aubry-Mather set is called a minimal foliation. In the case of
twist maps, it describes an invariant circle, while in solid state physics it
corresponds to a continuum of ground states. A Cantor Aubry-Mather set is
called a minimal lamination.
  In this paper we prove that when the rotation number of a minimal foliation
is either rational or easy to approximate by rational numbers, then the
foliation can be destroyed into a lamination by an arbitrarily small smooth
perturbation of the recurrence relation. This generalizes a theorem of Mather
for twist maps to general recurrence relations.
