Periodic points and measures for a class of skew-products

Abstract

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