Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size \begin{document}$ \delta $\end{document} , the physical measure varies continuously, with a modulus of continuity \begin{document}$ O(\delta \log \delta ) $\end{document} , which is asymptotically optimal for this kind of piecewise smooth maps.