Ergodic properties of infinite extensions of area-preserving flows

We consider volume-preserving flows (ᶠₜ) ₓ on S R, where S is a compact connected surface of genus g ≥ 2 and (ᶠₜ) ₓ has the form ᶠₜ (x, y) = (ₓx, y + ₀^{tf (ₒx) \, ds) } where (ₜ) ₓ 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 C^{2+ (S) }, then the following dynamical dichotomy holds: if there is a fixed point of (ₜ) ₓ on which f does not vanish, then (ᶠₜ) ₓ is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i. e. isomorphic to the trivial extension (⁰ₜ) ₓ. The proof of this result exploits the reduction of (ᶠₜ) ₓ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (ₜ) ₓ on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.