Periodic points and measures for a class of skew products

Abstract

We consider the open set constructed by M. Shub in [42] of partially hyperbolic skew products on the space T 2 ×T 2 whose non-wandering set is not stable. We show that there exists an open set U of such diffeomorphisms such that if F S ∈ 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 ∈ 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 R of U such that if F S ∈ 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.