A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus $\mathbb{T}^{2}$ there exists a $C^{1}$ open set $\mathcal{U}$ of $C^{1}$ regular maps such that every map belonging to $\mathcal{U}$ is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of $C^{1}$ toral diffeomorphisms, but that the property does hold for the class of $C^{1}$ diffeomorphisms on the 3-torus $\mathbb{T}^{3}$. Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of $C^{1}$ diffeomorphisms on the $n$-torus $\mathbb{T}^{n}$ ($n\ge 4$).