American Institute of Mathematical Sciences

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

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.
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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