Measurable rigidity of the Livsic equation for linear cocycles over hyperbolic systems

Measurable rigidity of the Livsic equation for linear cocycles over hyperbolic systems

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Clark Butler, Princeton University
Fine Hall 401

I will outline a proof that any measurable solution to the cohomological equation for Holder linear cocycles over a uniformly hyperbolic system coincides almost everywhere with a Holder solution. The focus will be on how one establishes uniform growth estimates for the cocycle from the existence of a measurable solution, which is the only obstruction to applying previously known results. This will be done by proving, more generally, that any Holder linear cocycle over a uniformly hyperbolic system which preserves a measurable inner product must also preserve a continuous inner product. I will also highlight some straightforward applications of this result to hyperbolic systems, including showing that if the geodesic flow on a closed negatively curved manifold has a measurably irreducible derivative action on its unstable bundle then the manifold has constant negative curvature.