Computable Condition for the Occurrence of Non-uniform Hyperbolicity in Families of One-dimensional Maps

can display a wide variety of dynamics which are closely intertwined. Graczyk-Świa̧teck proved that the set of regular parameters (corresponding to a hyperbolic periodic attractor) is open dense. Lyubich [8] proved that almost every parameter is either regular or stochastic (corresponding to an absolutely continuous invariant measure, acim for short). The second possibility in this dichotomy is not negligible, which had earlier been proved by Jakobson [5]. His theorem is recognized as a landmark in the study of chaotic dynamical systems, and so far quite a few different alternative proofs were given [1] [2] [6] [11] [12] [14] [15] [16] [17] [19]. However, any of these arguments tells nothing about the question: how many parameter values corresponding to acim in the quadratic family? We give a partial answer to this question. We develop a constructive argument closely following [6], and as a result obtain a lower estimate for the measure of the parameter set corresponding to acim near a = 2. Joint work with Stefano Luzzatto.