Linear cocycles over hyperbolic systems and criteria of conformality

Boris Kalinin; Victoria Sadovskaya

doi:10.3934/jmd.2010.4.419

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2010, Volume 4, Issue 3: 419-441. Doi: 10.3934/jmd.2010.4.419

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Linear cocycles over hyperbolic systems and criteria of conformality

Boris Kalinin^1, and
Victoria Sadovskaya^2,

1.
Department of Mathematics and statistics, University of South Alabama, Mobile, AL 36688
2.
Department of Mathematics and Statistics, ILB 325, University of South Alabama, Mobile, AL 36688

Received: July 31, 2009

Published: September 2010

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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.

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Mathematics Subject Classification: Primary: 37H15; Secondary: 37D20.

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