Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps
Stefano Luzzatto, Hiroki Takahasi
Abstract
We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurence 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 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 !
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