Monotone variational recurrence relations arise in solid state physics, conservative lattice dynamics and the theory of Hamiltonian twist maps. An example is the Frenkel-Kontorova lattice. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number ω ∈ ℝ. When ω is irrational, they are the action minimizers that constitute the Aubry-Mather set. This Aubry-Mather set is either connected or a Cantor set. In the first case it is called a minimal foliation and otherwise a minimal lamination. In this paper, we prove that when the rotation number of a minimal foliation is 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 a large class of monotone variational problems. We moreover provide a proof that may allow for further generalizations. © 2013 London Mathematical Society.