Ergodic properties of infinite extensions of area-preserving flows

Abstract

We consider volume-preserving flows $(Φ^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(Φ^f_t)_{t\in\mathbb{R}}$ has the form $Φ^f_t(x,y)=(ϕ_tx,y+\int_0^t f(ϕ_sx)ds)$, where $(ϕ_t)_{t\in\mathbb{R}}$ 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 $\mathscr{C}^{2+ε}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(ϕ_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, then $(Φ^f_t)_{t\in\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(Φ^0_t)_{t\in\mathbb{R}}$. The proof of this result exploits the reduction of $(Φ^f_t)_{t\in\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(ϕ_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.