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The twist map, the extended Frenkel-Kontorova model and the devil’s staircase. (English) Zbl 0559.58013
Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica 7D, 240-258 (1983).
[For the entire collection see Zbl 0536.00007.]
The paper reviews exact results on the extended discrete Frenkel- Kontorova (FK) model and the associated area preserving twist maps of the cylinder onto itself. The structure of the ground states in the FK model is studied. For a fixed irrational rotation number the union of all the ground states (called incommensurate in this case) is either a KAM circle or, beyond a transition parameter, becomes a minimal Cantor set with zero length measure. A series of rigorous upper bounds for these breaking of the KAM curve parameters is obtained.
For a rational rotation number the periodic ground states (called commensurate) are studied. The existence of heteroclinic orbits (discommensurate states) is asserted. The Peierls-Nabarro barrier is studied in this situation. Necessary conditions for the trajectories to represent metastable configurations, which can be chaotic, are given. For the FK model it is asserted that the atomic mean distance versus a chemical potential is a Cantor function (the devil’s staircase).
Independently a similar theory was recently developed by J. Mather, see the survey paper by A. Chenciner [Astérisque 121/122, 147-170 (1985)].
Reviewer: F.Przytycki

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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