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

We consider diffeomorphisms in $\overline {\mathcal {O}}^\infty (\mathbb {T}^2)$, the $C^\infty$-closure of the conjugancy class of translations of $\mathbb {T}^2$. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-type property, which takes into account the “directions” of mixing, and we prove that generic elements of $\overline {\mathcal {O}}^\infty (\mathbb {T}^2)$ satisfy this property. As a consequence, we obtain a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.