A quantitative, computer assisted, version of Jakobson's theorem on the occurrence of stochastic dynamics in one-dimensional dyn

We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurrence of nonuniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we obtain a first ever explicit lower bound for the set of parameters corresponding to maps in the quadratic family f_{a}(x) = x^{2}-a which have an absolutely continuous invariant probability measure.