Jakobson's theorem asserts that the set of real $c$ such that there is a measure on $\mathbb R$ absolutely continuous with respect to Lebesgue measure and invariant by the polynomial $x^2+c$ has positive measure.

This theorem has puzzled many mathematicians, and many (Guckenheimer, Carleson, Benedicks, Yoccoz) have written their own proof. Unfortunately, all these proofs are very difficult to read. I will offer a new proof, more combinatorial. I hope that it is easier to understand.