Ergodic properties of infinite extensions of area-preserving flows
Abstract
We consider volume-preserving flows (ft)t∈R on S× R, where S is a closed connected surface of genus g≥ 2 and (ft)t∈R has the form ft(x,y)=(φtx,y+∫0t f(φsx)ds), where (φt)t∈R 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∈R on which f does not vanish, then (ft)t∈R is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (0t)t∈R. The proof of this result exploits the reduction of (ft)t∈R to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φt)t∈R on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.