PERIODIC POINTS AND MEASURES FOR A CLASS OF SKEW-PRODUCTS

We consider the C-1-open set V of partially hyperbolic diffeomorphisms on the space T-2 x T-2 whose non-wandering set is not stable, introduced by M. Shah in [5]. Firstly, we show that the non-wandering set of each diffeormorphism in V is a limit of horseshoes in the sense of entropy. Afterwards, we establish the existence of a C-2-open set U of C-2-diffeomorphisms in V and of a C-2-residual subset R of U such that any diffeomorphism in 91 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 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.