  For a class of robustly transitive diffeomorphisms on $\mathbb T^4$
introduced by Shub in [24], satisfying an additional bunching condition, we
show that there exits a $C^2$ open and $C^r$ dense subset $\mathcal U^r$,
$2\leq r\leq\infty$, such that any two hyperbolic points of $g\in \mathcal U^r$
with stable index $2$ are homoclinically related. As a consequence, every $g\in
\mathcal U^r$ admits a unique homoclinic class associated to the hyperbolic
periodic points with index $2$, and this homoclinic class coincides to the
whole ambient manifold. Moreover, every $g\in \mathcal U^r$ admits at most one
measure with maximal entropy, and every $g\in\mathcal U^{\infty}$ admits a
unique measure of maximal entropy.

Dynamical Systems