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.