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

Abstract

<p style='text-indent:20px;'>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). <p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \delta $\end{document}</tex-math></inline-formula>, the physical measure varies continuously, with a modulus of continuity <inline-formula><tex-math id="M2">\begin{document}$ O(\delta \log \delta ) $\end{document}</tex-math></inline-formula>, which is asymptotically optimal for this kind of piecewise smooth maps.