In its geometric setting the Aubry-Mather Theory studies minimal geodesics in compact Riemannian manifolds, in particular in Riemannian tori. The present proposal aims at developing an analogous theory for holomorphic maps from the complex plane into a tame almost complex torus. These complex lines should generalize the affine complex lines in a standard complex torus. Although the nature of the differential equation and in particular the absence of a variational characterization of complex lines make the two problems very different there are some striking analogies: A KAM type perturbation result by J. Moser and global results by the proposer.
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