Periodic points and measures for a class of skew-products

We consider the C1-open set $$\cal{V}$$ of partially hyperbolic diffeomorphisms on space $$\mathbb{T}^{2}\times\mathbb{T}^{2}$$ whose non-wandering is not stable, introduced by M. Shub in [57]. Firstly, we show that each diffeormorphism a limit horseshoes sense entropy. Afterwards, establish existence C2-open $$\cal{U}$$ C2-diffeomorphisms and C2-residual subset ℜ such any diffeomorphism has equal topological periodic entropies, asymptotic per-expansive, sub-exponential growth rate orbits admits principal strongly faithful symbolic extension with embedding. Besides, unique probability measure maximal entropy describing distribution orbits. Under an additional assumption, prove skew-products preserve ergodic SRB measure, which physical, basin full Lebesgue coincides