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We consider transformations preserving a contracting foliation, such that theassociated quotient map satisfies a Lasota-Yorke inequality. We prove that theassociated transfer operator, acting on suitable normed spaces, has a spectralgap (on which we have quantitative estimation). As an application we considerLorenz-like two dimensional maps (piecewise hyperbolic with unboundedcontraction and expansion rate): we prove that those systems have a spectralgap and we show a quantitative estimate for their statistical stability. Underdeterministic perturbations of the system of size $\delta$, the physicalmeasure varies continuously, with a modulus of continuity $O(\delta \log \delta)$, which is asymptotically optimal for this kind of piecewise smooth maps.

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