On the destruction of minimal foliations

TL;DR

This paper proves that minimal foliations in monotone variational recurrence relations, including physical models like the Frenkel-Kontorova lattice, can be destroyed into laminations with small smooth perturbations, extending Mather's theorem.

Contribution

It generalizes Mather's theorem from twist maps to broader recurrence relations, showing minimal foliations can be destroyed into laminations via small perturbations.

Findings

Minimal foliations can be destroyed into laminations with small smooth perturbations.

The result applies to irrational and rational rotation numbers, including those well-approximated by rationals.

Extends Mather's theorem from twist maps to general recurrence relations.

Abstract

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.