Ergodic properties of skew products in infinite measure

Let (a"broken vertical bar, A mu) be a shift of finite type with a Markov probability, and (Y, nu) a non-atomic standard measure space. For each symbol i of the symbolic space, let I broken vertical bar (i) be a non-singular automorphism of (Y, nu). We study skew products of the form (omega, y) a dagger broken vertical bar (sigma omega, I broken vertical bar(omega 0) (y)), where sigma is the shift map on (a"broken vertical bar, A mu). We prove that, when the skew product is recurrent, it is ergodic if and only if the I broken vertical bar (i) 's have no common non-trivial invariant set. In the second part we study the skew product when a"broken vertical bar = {0, 1}(Z), A mu is a Bernoulli measure, and I broken vertical bar(0),I broken vertical bar(1) are R-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to , and its trajectories satisfy the central, functional central and local limit theorem.