Stable ergodic properties of cocycles over hyperbolic attractors (Q1809225)

The author considers the extension \(\Phi^t\) of a hyperbolic flow \(\Psi^t\)~ on a hyperbolic attractor \(\Lambda\) by a torus \(T^n\) which is defined by an \(n\)-tuple \((f_1,\dots,f_n)\) of Hölder continuous functions on \(\Lambda\) via \[ \Phi^t(x,y_1,\dots,y_n)= (\Psi^t x,e^{2\pi i\int^t_0 f_1(\Psi^u x)du} y_{1,\dots,}e^{2\pi i\int^t_0 f_n(\Psi^ux)du}y_n). \] Every \(\Psi^t\)-invariant Borel-measure \(\mu\) on \(\Lambda\) can be combined with the Haar measure on \(T^n\) to a \(\Phi^t\)-invariant measure on \(\Lambda\times T^n\). If \(\mu\) is the Gibbs equilibrium state of a Hölder function, then it is shown that the set of tuples \((f_1,\dots,f_n)\) for which \(\Phi^t\) is ergodic is open and dense with respect to the Hölder topology. A similar result holds for every compact Lie group extension of \(\Psi^t\). If \(\Psi^t\) is mixing, then an open dense set of extensions of \(\Psi^t\) is mixing as well. As an application, the author rediscovers a result of Brin: There is an open dense set of smooth negatively curved metrics on a given closed manifold for which the frame flow is mixing.