Poincaré-Birkhoff theorems in random dynamics

The Poincar\'e-Birkhoff Theorem states that an area-preserving periodic twist map R x [-1,1] -> R x [-1,1] has two geometrically distinct fixed points. We generalize it to area-preserving twist maps F : (R x [-1,1]) x \Omega -> R x [-1,1] that are random with respect to a \tau-invariant ergodic probability measure P on a separable metric space \Omega, where \tau is a continuous R-action on \Omega. We will prove, in particular, that the probability that the area-preserving twist F(-,-,;\omega), \omega \in \Omega, has fixed points, is one. The proofs are based on the notion of "random generating function", and we use a calculus suited for their study.