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

TL;DR

This paper introduces a new mixing-like property for minimal diffeomorphisms of the 2-torus, demonstrating that generically such maps lack invariant foliations and satisfy this property, with results applicable in analytic settings.

Contribution

It defines a novel mixing-like property for minimal diffeomorphisms and proves that generic maps in this class lack invariant foliations, extending understanding of their dynamical complexity.

Findings

Generic diffeomorphisms are minimal and uniquely ergodic.

Most such diffeomorphisms satisfy the new mixing-like property.

There exists a residual set of diffeomorphisms without invariant foliations.

Abstract

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.