Spectral Gap for Contracting Fiber Systems and Applications

We consider transformations preserving a contracting foliation, such that the associated quotient map satis es a Lasota Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces,has spectral gap. As an application we consider Lorenz-Like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have spectral gap and we show a quantitative estimation for their statistical stability. Under deterministic perturbations of the system, the physical measure varies continuously, with a modulus of continuity O(delta log (delta) ).