  This work is addressed to study Anosov endomorphisms of $\mathbb{T}^d,$
$d\geq 3.$ We are interested to obtain metric and topological information on
such Anosov endomorphism by comparison between their Lyapunov exponents and the
ones of its linearization. We can characterize when a weak unstable foliation
of a special Anosov endomorphism near to linear is an absolutely continuous
foliation. Also, we show that in dimension $d \geq 3,$ it is possible to find a
smooth special Anosov endomorphism being conservative but not Lipschitz
conjugate with its linearization, in contrast with the smooth rigidity in
dimension two.

Dynamical Systems