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We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

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Statistical stability and linear response for random hyperbolic dynamics

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

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2 results

Statistical stability and linear response for random hyperbolic dynamics

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts