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 d x d t = − ∇ 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.
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