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Geodesic flows on the two-dimensional torus and phase transitions “commensurability-noncommensurability”. (English. Russian original) Zbl 0641.58032
Funct. Anal. Appl. 20, 260-266 (1986); translation from Funkts. Anal. Prilozh. 20, No. 4, 9-16 (1986).
The authors consider invariant sets of geodesic flows on the torus \(T^2\). For each rational member of rotation, they construct such an invariant set being torus or Cantor torus. The main result is that an invariant Lagrangian torus of geodesic flows has projection on the base of cotangent bundle if and only if its geodesic trajectories are A-geodesic (minimizing length). Aubry’s method is adapted for the geodesic flows on \(T^2\).
Reviewer: P.Khmelevskaya

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
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