Solutions to the twisted cocycle equation over hyperbolic systems

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October 2000, 6(4): 935-946. doi: 10.3934/dcds.2000.6.935

Solutions to the twisted cocycle equation over hyperbolic systems

1.	Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

Received February 2000 Revised May 2000 Published August 2000

Abstract
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A twisted cocyle with values in a Lie group $G$ is a cocyle that incorporates an automorphism of $G$. Suppose that the underlying transformation is hyperbolic. We prove that if two Hölder continuous twisted cocycles with a sufficiently high Hölder exponent assign equal 'weights' to the periodic orbits of $\phi$, then they are Hölder cohomologous. This generalises a well-known theorem due to Livšic in the untwisted case. Having determined conditions for there to be a solution to the twisted cocycle equation, we consider how many other solution there may be. When $G$ is a toius, we determine conditions for there to be only finitely many solutions to the twisted cocycle equation.

Keywords: cocycle equations., Hyperbolic dynamics.

Mathematics Subject Classification: 58F1.

Citation: C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935

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