ERGODIC PROPERTIES OF SKEW PRODUCTS WITH LASOTA-YORKE TYPE MAPS IN THE BASE

We consider skew products T(x, y) = (f(x), T(e)(x) y) preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and T(i), i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T. We apply these results to random perturbations.