Non-integrability of dominated splitting on $\mathbb{T}^2$
  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. 
Baolin He, and Shaobo Gan
math.DS