A Proof of Jakobson’s Theorem via Yoccoz puzzles and the measure of stochastic parameters

Jakobson's theorem says that a certain family of unimodal maps of the interval has a positive measure set of ``stochastic'' parameters for which there exist invariant measure absolutely continuous with respect to the Lebesgue measure. Luzzatto-Takahasi gave an effective estimate on the measure, but it was like $10^{-5000}$ for the quadratic family. We will present an alternative approach to Jakobson's theorem using complex extension and Yoccoz puzzle/parapuzzle techniques, and try to improve the estimates on the measure of stochastic parameters.