The Aubry-Mather theorem for driven generalized elastic chains.

*(English)*Zbl 1293.34017This 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.

Reviewer: Caidi Zhao (Wenzhou)

##### 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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\textit{S. Slijepčević}, Discrete Contin. Dyn. Syst. 34, No. 7, 2983--3011 (2014; Zbl 1293.34017)

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