  We consider the open set constructed by M. Shub in [42] of partially
hyperbolic skew products on the space $\mathbb{T}^2\times \mathbb{T}^2$ whose
non-wandering set is not stable. We show that there exists an open set
$\mathcal{U}$ of such diffeomorphisms such that if $F_S\in \mathcal{U}$ then
its measure of maximal entropy is unique, hyperbolic and, generically,
describes the distribution of periodic points. Moreover, the non-wandering set
of such an $F_S\in \mathcal{U}$ contains closed invariant subsets carrying
entropy arbitrarily close to the topological entropy of $F_S$ and within which
the dynamics is conjugate to a subshift of finite type. Under an additional
assumption on the base dynamics, we verify that $F_S$ preserves a unique SRB
measure, which is physical, whose basin has full Lebesgue measure and coincides
with the measure of maximal entropy. We also prove that there exists a residual
subset $\mathcal{R}$ of $\mathcal{U}$ such that if $F_S\in \mathcal{R}$ then
the topological and periodic entropies of $F_S$ are equal, $F_S$ is asymptotic
per-expansive, has a sub-exponential growth rate of the periodic orbits and
admits a principal strongly faithful symbolic extension with embedding.
