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### Open Access Journals

DCDS

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.

DCDS

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.

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