We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly 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 $zeta$-Holder 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 $delta$, we show that the physical measure varies continuously with respect to a strong $L^infty$-like norm. Moreover, we prove that for certain interesting classes of perturbations its modulus of continuity is $O(delta^zeta log delta)$.
We consider a robust class of random non-uniformly expanding local homeomorphisms and Holder continuous potentials with small variation. For each element of this class we develop the Thermodynamical Formalism and prove the existence and uniqueness of
equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.
