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We consider diffeomorphisms in the $C^\infty$-closure of the conjugancy classof translations of the 2-torus. By a theorem of Fathi and Herman, a genericdiffeomorphism in that space is minimal and uniquely ergodic. We define a newmixing-like property, which takes into account the "directions" of mixing, andwe prove that generic elements of the space in question satisfy this property.As a consequence, we show that there is a residual set of strictly ergodicdiffeomorphisms without invariant foliations of any kind. We also obtain ananalytic version of these results.

A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus