Ergodic properties of skew products in infinite measure

Abstract

Let (\Omega,\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 \Phi_i be a measure-preserving automorphism of (Y,\nu). We study skew products of the form (\omega,y) --> (\sigma\omega,\Phi_{\omega_0}(y)), where \sigma =shift map on (\Omega,\mu). We prove that, when the skew product is conservative, it is ergodic if and only if the \Phi_i's have no common non-trivial invariant set. In the second part we study the skew product when \Omega={0,1}^Z, \mu =Bernoulli measure, and \Phi_0,\Phi_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 \sqrt{n}, and its trajectories satisfy the central, functional central and local limit theorem.