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

On Lyapunov exponents properties of special Anosov endomorphisms on $\mathbb{T}^d$