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The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states. (English) Zbl 1237.37059
Summary: We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of ground-states is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned.

37N05 Dynamical systems in classical and celestial mechanics
37J50 Action-minimizing orbits and measures (MSC2010)
37E40 Dynamical aspects of twist maps
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