Uniform Hyperbolicity on Random Sets

Let Λ be a random compact invariant (with respect to a random map) set with a uniform tangent bundle splitting. We show that if there is a random non-uniformly hyperbolic variable x0 on Λ and the space of limit points of time averages of random Dirac measures on the orbit of x0 coincides with the space of invariant probability measures supported on Λ then, Λ is a random uniformly hyperbolic set.