Random conservative cylinder maps

A theorem of Kakutani gives necessary and sufficient conditions for ergodicity of a probability measure-preserving random dynamical system. If the condition of finiteness of the measure is dropped, then the result is no longer true.
There is, however, a large class of conservative maps of the cylinders, given by R-extensions of a same probability measure-preserving transformation, where the theorem remains true. We will discuss this class and prove that, in this context, Kakutani's theorem indeed remains valid. If time permits, we also discuss the statistical properties of the Birkhoff sums. Joint work with Patricia Cirilo and Enrique Pujals.