engleski

The Aubry-Mather theorem for driven generalized elastic chains

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 space-time invariant measures, denoted by $\mathcal{; ; ; A}; ; ; $, projects injectively to a dynamical system on a 2-dimensional cylinder. We also prove existence of space-time ergodic 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, A consists entirely of equilibria and uniformly sliding solutions. The key tool is a new weak Lyapunov function on the space of translationally nvariant probability measures on the state space, which counts intersections.

Aubry-Mather theory ; Frenkel-Kontorova model ; twist maps ; attractors ; Poincar\'{; ; ; e}; ; ; -Bendixson theorem ; reaction-diffusion equation ; synchronization ; uniformly sliding states ; minimizing measures ; space-time invariant measure ; space-time entropy

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