Stefano Luzzatto (Imperial College, London, UK)

25 years on: a quantitative, computer-assisted, version of Jakobson's Theorem.

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 an application of our results we obtain a first ever explicit lower bound for the measure of 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.

This is joint work with H. Takahasi (Kyoto University).