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arXiv · 1510.07719

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

Clark Butler
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Abstract

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of $\mathbb{R}^{d}$. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.

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Clark Butler. 2018-07-23. Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems. https://arxiv.org/abs/1510.07719

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