  We found a dichotomy involving the unstable Lyapunov exponent of a special
Anosov endomorphism of the torus induced by the conjugacy with the
linearization. In fact, either every unstable leaf meets on a set of zero
measure the set for which is defined such unstable Lyapunov exponent or the
endomorphims is smoothly conjugated with its linearization. Also we are able to
characterize the absolute continuity of the intermediate foliation for a class
of volume preserving special Anosov endomorphisms of $\mathbb{T}^3.$
