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

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

2009-02-14

arxiv.org/abs/0902.2474v1

Proc. Amer. Math. Soc. 137 (2009), pp. 3379-3386

Mathematics

Dynamical Systems

8 pages, to appear in Proc. Amer. Math. Soc

Scientific paper

10.1090/S0002-9939-09-09903-1

We consider diffeomorphisms in the $C^\infty$-closure of the conjugancy class of translations of the 2-torus. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-like property, which takes into account the "directions" of mixing, and we 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 ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.

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