  We consider uniformly (DC) or periodically (AC) driven generalized infinite
elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics.
We first show that the union of supports of all the invariant measures, denoted
by A, projects injectively to a dynamical system on a 2-dimensional cylinder.
We also prove existence of ergodic invariant measures supported on a set of
rotationaly ordered configurations with an arbitrary (rational or irrational)
rotation number. This shows that the Aubry-Mather structure of ground states
persists if an arbitrary AC or DC force is applied. The set A attracts almost
surely (in probability) configurations with bounded spacing. In the DC case,
the set A consists entirely of equilibria and uniformly sliding solutions. The
key tool is a new weak Lyapunov function on the space of translationally
invariant probability measures on the state space, which counts intersections.

Dynamical Systems