  Let $f$ be a non-invertible irreducible Anosov map on $d$-torus. We show that
if the stable bundle of $f$ is one-dimensional, then $f$ has the integrable
unstable bundle, if and only if, every periodic point of $f$ admits the same
Lyapunov exponent on the stable bundle with its linearization. For
higher-dimensional stable bundle case, we get the same result on the assumption
that $f$ is a $C^1$-perturbation of a linear Anosov map with real simple
Lyapunov spectrum on the stable bundle. In both cases, this implies if $f$ is
topologically conjugate to its linearization, then the conjugacy is smooth on
the stable bundle.
