The Aubry-Mather theorem for driven generalized elastic chains

Abstract

We consider uniformly (DC) or periodically (AC) driven generalizedinfinite elastic chains (a generalized Frenkel-Kontorova model) withgradient dynamics. We first show that the union of supports of allspace-time invariant measures, denoted by $\mathcal{A}$, projectsinjectively to a dynamical system on a 2-dimensional cylinder. We also proveexistence of space-time ergodic measures supported on a set of rotationalyordered configurations with an arbitrary (rational or irrational) rotationnumber. This shows that the Aubry-Mather structure of ground states persistsif an arbitrary AC or DC force is applied. The set $\mathcal{A}$ attractsalmost surely (in probability) configurations with bounded spacing. In theDC case, $\mathcal{A}$ consists entirely of equilibria and uniformly slidingsolutions. The key tool is a new weak Lyapunov function on the space oftranslationally invariant probability measures on the state space, whichcounts intersections.