We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We prove that the associated transfer operator, acting on suitable anisotropic normed spaces, has a spectral gap (on which we have quantitative estimation) and the disintegration of the unique invariant physical measure, along the stable leaves, is ζ-Hölder. We use this fact to obtain exponential decay of correlations on the set of ζ-Hölder functions.

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic dynamics semi-conjugated to nonuniformly expanding maps. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We apply the spectral gap property and the ζ-Hölder regularity of the disintegration of its physical measure to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size δ, we show that the physical measure varies continuously with respect to a strong L ∞ -like norm. Moreover, we prove that for certain interesting classes of perturbations its modulus of continuity is O(δ ζ log δ).