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We consider a general relation between fixed point stability of suitablyperturbed transfer operators and convergence to equilibrium (a notion which isstrictly related to decay of correlations). We apply this relation todeterministic perturbations of a class of (piecewise) partially hyperbolic skewproducts whose behavior on the preserved fibration is dominated by theexpansion of the base map. In particular we apply the results to power lawmixing toral extensions. It turns out that in this case, the dependence of thephysical measure on small deterministic perturbations, in a suitableanisotropic metric is at least Holder continuous, with an exponent which isexplicitly estimated depending on the arithmetical properties of the system. Weshow explicit examples of toral extensions having actually Holder stability andnon differentiable dependence of the physical measure on perturbations.

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