We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.
Homoclinic classes of geodesic flows on rank 1 manifolds
Holography of geodesic flows, harmonizing metrics, and billiards' dynamics
Geodesic flows and the mother of all continued fractions
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