Ergodic properties of infinite extensions of area-preserving flows

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Corinna Ulcigrai, University of Bristol

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 SxR given by the solutions to the differential equations dx/dt=X, dy/dt=f, where (x,y) are coordinates on SxR 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.