Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products

We consider a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a class of (piecewise) partially hyperbolic skew products whose behavior on the preserved fibration is dominated by the expansion of the base map. In particular, we apply the results to power law mixing toral extensions. It turns out that in this case, the dependence of the physical measure on small deterministic perturbations, in a suitable anisotropic metric, is at least Hölder continuous, with an exponent which is explicitly estimated depending on the arithmetical properties of the system. We show explicit examples of toral extensions having actually Hölder stability and non differentiable dependence of the physical measure on perturbations.