Solutions to the twisted cocycle equation over hyperbolic systems

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.