Non-integrability of dominated splitting on ${\mathbb{T}}^2$

Abstract

We construct a diffeomorphism $f$ on 2-torus with a dominated splitting $E\oplus F$ such that there exists an open neighborhood $\mathcal{U}\ni f$ satisfying that for any $g\in \mathcal{U}$, neither $E_g$ nor $F_g$ is integrable.