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In this paper we study Holder continuous linear cocycles over transitiveAnosov diffeomorphisms. Under various conditions of relative pinching weestablish properties including existence and continuity of measurable invariantsub-bundles and conformal structures. We use these results to obtain criteriafor cocycles to be isometric or conformal in terms of their periodic data. Weshow that if the return maps at the periodic points are, in a sense, conformalor isometric then so is the cocycle itself with respect to a Holder continuousRiemannian metric.

Linear cocycles over hyperbolic systems and criteria of conformality