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The Aubry-Mather theorem for driven generalized elastic chains. (English) Zbl 1293.34017
This paper studies uniformly (DC) or periodically (AC) driven generalized infinite elastic chains with gradient dynamics. The authors first show that the union of supports of all space-time invariant measures, denoted by \(\mathcal{A}\), projects injectively to a dynamical system on a two dimensional cylinder. They also prove the existence of space-time ergodic measures supported on a set of rotationally ordered configurations with an arbitrary rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The set \(\mathcal{A}\) attracts almost surely in probability configurations with bounded spacing. In the DC case, \(\mathcal{A}\) 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.

MSC:
34A33 Ordinary lattice differential equations
34C12 Monotone systems involving ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37E40 Dynamical aspects of twist maps
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