  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 Holder 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 Holder stability and
non differentiable dependence of the physical measure on perturbations.
