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Linear cocycles and their periodic data

4:00 pm Thursday, October 7, 2010
Boris Kalinin (University of South Alabama)

A linear cocycle over a dynamical system ƒ :  M → M is an automorphism of a vector bundle over M that projects to ƒ. An important example comes from the differential Dƒ or its restriction to an invariant sub-bundle of TM. For a trivial bundle with fiber Rd, a linear cocycle can be simply viewed as a GL(d,R)-valued function on the manifold. We consider Hoelder continuous linear cocycles over hyperbolic systems. In this case it is natural to look at the behavior of a cocycle at the periodic orbits of the system, which we call the periodic data. We discuss what conclusions can be made about the cocycle based on its periodic data. In particular, we obtain criteria for a cocycle to be isometric or conformal.

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