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

Stefano Galatolo, Rafael Lucena

Published 2015-07-29Version 1

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 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 $O(\delta \log \delta ).$