Ghost circles in lattice Aubry–Mather theory

Abstract

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps.Such recurrence relations often admit a variational structure, so that the solutions x : Z d → R are the stationary points of a formal action function W (x). Given any rotation vector ω ∈ R d , classical Aubry-Mather theory establishes the existence of a large collection of solutions of ∇W (x) = 0 of rotation vector ω. For irrational ω, this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps.In this paper, we study the parabolic gradient flow dx dt = −∇W (x) and we will prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called 'ghost circle'. The existence of these ghost circles is known in dimension d = 1, for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions.As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.

On the destruction of minimal foliations

“…We would say that our proof is a bit simpler. We are moreover confident that our proof allows for a generalization to variational problems on lattices, such as those discussed in [6], [10] or [23].…”

“…For the proofs of our statements we refer to [23]. In one form or another, much of this section can also be found in the standard references [2], [8], [14] and [17].…”

“…This means that there are x − , x + ∈ M ω so that x − ≪ x ≪ x + , but that there is no y ∈ M ω with x − ≪ y ≪ x + . By a standard result, see for instance [3], [19] or [23], the gap […”

“…Or, putting it more precisely, if the map (x, y) → (x, X) = (x, T1(x, y)) is a global orientation preserving diffeomorphism of R 2 . One can show, see [8] or [23], that a Hamiltonian twist map admits a so-called generating function S = S(x, X) from R × R into R with the property that a sequence {. .…”

“…In order not to overload the reader at this point, we will discuss some more classical theory later, in Section 2. For a much more complete overview of Aubry-Mather theory, we refer to [17] or [23].…”