Shub's example revisited

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 ${\mathcal{U}}^r$, $2\le r\le \mathrm{\infty }$, 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 $g\in {\mathcal{U}}^{\mathrm{\infty }}$ admits a unique measure of maximal entropy.