Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

Abstract

We formulate and prove a Jakobson–Benedicks–Carleson-type theorem on the occurrence of non-uniform 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 show that the set of parameters corresponding to maps in the quadratic family fa(x) = x2 − a which have an absolutely continuous invariant probability measure is at least 10−5000.