# Ergodic properties of infinite extensions of area-preserving flows

# Ergodic properties of infinite extensions of area-preserving flows

We consider infinite volume preserving flows that are obtained as extensions of flows on surfaces. Consider a smooth area-preserving flow on a surface $S$ given by a vector field $X$ and consider a real valued function $f$ on $X$. The (skew-product) extension of the flow on $S$ given by $f$ is the flow on $S x R$ given by the solutions to the differential equations $dx/dt=X$, $dy/dt=f$, where $(x,y)$ are coordinates on $S x R$ and $R$ is the real line. While the ergodic properties of surface flows that preserve a smooth area form are now well understood (as we will summarize), very little is known for these infinite measure preserving extensions, which were previously studied only when S is a torus. We prove ergodicity of the infinite extension when the surface flow is of periodic type and f belong to a suitable subspace of smooth functions. In the proof we develop renormalization techniques for cocycles with logarithmic singularities over interval exchange transformations. This is joint work with K. Fraczek.