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Mather sets for twist maps and geodesics on tori. (English) Zbl 0664.53021

Dyn. Rep. 1, 1-56 (1988).
[For the entire collection see Zbl 0651.00018.]
This paper is an excellent survey on independent researches in different fields as differential geometry, dynamical systems and solid state physics, and more precisely on: (1) geodesics on a 2-dimensional torus with Riemannian or symmetric Finsler metric; (2) the dynamics of monotone twist maps of an annulus; (3) the discrete Frenkel-Kontorova model.
The author introduces a variational problem which can be interpreted in each of the above cases so that it becomes a common root of the proofs. He studies the minimal trajectories of the variational principle and specializes the results to the three situations. In the case (1) he gives a precise description of the set of minimal geodesics on the torus, i.e. geodesics which - when considered on the universal cover - minimize arclength between any two of their points. He presents and extends results by G. K. Hedlund [Ann. Math., II. Ser. 33, 719-739 (1932; Zbl 0006.32601)] and H. M. Morse [Trans. Am. Math. Soc. 26, 25-60 (1924; JFM 50.0466.04)] and he proves some new results which complete the work of Hedlund and Morse to a certain degree. There are also examples of Riemannian metrics on tori which display the two extremal possibilities for the set of minimal geodesics, as the integrable geodesic flows and several classes of metrics for which the set is small in various respects.
For the case (2), the author studies area-preserving monotone twist maps \(\phi: S^ 1\times [0,1]\to S^ 1\times [0,1]\) preserving the boundary components. Mather sets are particular \(\phi\)-invariant subset of \(S^ 1\times [0,1]\). There are alternative proofs for the results obtained by J. N. Mather [Topology 21, 457-467 (1982; Zbl 0506.58032)], and examples closely related to the ones given by Mather. In the case (3), the author describes the minimum energy configurations of the discrete Frenkel-Kontorova model. He obtains the same results proved by S. Aubry and P. Y. LeDaeron [Physica D 8, 381-422 (1983)], with different proofs and under slightly weaker hypothesis. Relations and differences between the problems in (1), (2), (3) are pointed out and a guide to some of the recent literature is given.
Reviewer: A.M.Pastore

MSC:

53C22 Geodesics in global differential geometry
58E20 Harmonic maps, etc.
37Cxx Smooth dynamical systems: general theory
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
70Gxx General models, approaches, and methods in mechanics of particles and systems