Stable topological transitivity properties of ℝ n -extensions of hyperbolic transformations

We consider ℝ n skew-products of a class of hyperbolic dynamical systems. It was proved by Niţică and Pollicott [Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269] that for an Anosov diffeomorphism ϕ of an infranilmanifold Λ there is (subject to avoiding natural obstructions) an open and dense set f :Λ→ℝ N for which the skew-product ϕ f ( x , v )=(ϕ( x ), v + f ( x )) on Λ×ℝ N has a dense orbit. We prove a similar result in the context of an Axiom A hyperbolic flow on an attractor.