Stable ergodicity of skew products of one-dimensional hyperbolic flows
C.P. Walkden
Discrete & Continuous Dynamical Systems - A 1999, 5(4): 897-904 doi: 10.3934/dcds.1999.5.897
We consider hyperbolic flows on one dimensional basic sets. Any such flow is conjugate to a suspension of a shift of finite type. We consider compact Lie group skew-products of such symbolic flows and prove that they are stably ergodic and stably mixing, within certain naturally defined function spaces.
keywords: skew product symbolic flow Hyperbolic flow stable ergodicity.
Solutions to the twisted cocycle equation over hyperbolic systems
C.P. Walkden
Discrete & Continuous Dynamical Systems - A 2000, 6(4): 935-946 doi: 10.3934/dcds.2000.6.935
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

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