  In this paper we focused our study on Derived From Anosov diffeomorphisms (DA
diffeomorphisms ) of the torus $\mathbb{T}^3,$ it is, an absolute partially
hyperbolic diffeomorphism on $\mathbb{T}^3$ homotopic to an Anosov linear
automorphism of the $\mathbb{T}^3.$ We can prove that if $f: \mathbb{T}^3
\rightarrow \mathbb{T}^3 $ is a volume preserving DA diffeomorphism homotopic
to linear Anosov $A,$ such that the center Lyapunov exponent satisfies
$\lambda^c_f(x) > \lambda^c_A > 0,$ with $x $ belongs to a positive volume set,
then the center foliation of $f$ is non absolutely continuous. We construct a
new open class $U$ of non Anosov and volume preserving DA diffeomorphisms,
satisfying the property $\lambda^c_f(x) > \lambda^c_A > 0$ for $m-$almost
everywhere $x \in \mathbb{T}^3.$ Particularly for every $f \in U,$ the center
foliation of $f$ is non absolutely continuous.
