Ergodic properties of infinite extensions of area-preserving flows

We consider volume-preserving flows (Φᶠₜ)ₜ on S × R , where S is a compact connected surface of genus g ≥ 2 and (Φᶠₜ)ₜ has the form Φᶠₜ(x, y) = (φₜx, y + ∫₀ᵗf(φₛx)\,ds) where (φₜ)ₜ is a locally Hamiltonian flow of hyperbolic periodic type on S and f is a smooth real valued function on S. We investigate ergodic properties of these infinite measure-preserving flows and prove that if f belongs to a space of finite codimension in C2+ε(S) , then the following dynamical dichotomy holds: if there is a fixed point of (φₜ)ₜ on which f does not vanish, then (Φᶠₜ)ₜ is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (Φ⁰ₜ)ₜ . The proof of this result exploits the reduction of (Φᶠₜ)ₜ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φₜ)ₜ on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.