Abstract
In the present paper, we study the distribution of the return points in the fibers for a random nonuniformly expanding dynamical system, preserving an ergodic probability. We also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property among fibers. As consequences, we prove that any expanding measure is the limit of probability measures whose measures of disintegration on the fibers are supported on a finite number of return points and we prove that the average of the measures on the fibers corresponding to a disintegration, along the orbit (θn(w))n≥0 in the base dynamics is the limit of Dirac measures supported on return orbits on the fibers.
