Journal of Modern Dynamics (JMD)

Linear cocycles over hyperbolic systems and criteria of conformality

Pages: 419 - 441, Issue 3, July 2010      doi:10.3934/jmd.2010.4.419

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Boris Kalinin - Department of Mathematics and statistics, University of South Alabama, Mobile, AL 36688, United States (email)
Victoria Sadovskaya - Department of Mathematics and Statistics, ILB 325, University of South Alabama, Mobile, AL 36688, United States (email)

Abstract: In this paper, we study Hölder-continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant subbundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a Hölder-continuous Riemannian metric.

Keywords:  cocycle, hyperbolic system, periodic data, pinching, conformality, isometry.
Mathematics Subject Classification:  Primary: 37H15; Secondary: 37D20.

Received: August 2009;      Available Online: October 2010.