Abundance of unimodal maps with dense critical orbit and prefixed critical orbit (Q1578233)

Consider the quadratic family \(f_a(x) = 1 - a x^2\) on the interval \([-1,1]\), depending on a parameter \(a\). M. Jakobson showed that for a set \(E\) of parameter values of positive Lebesgue measure and with \(a = 2\) as a one-sided density point, \(f_a(x)\) admits an absolutely continuous invariant measure. A different proof was given by M. Benedicks and L. Carleson. Following their work, the authors work out the argument for the fact that the orbit of the critical point lies dense in \([f^2_a(0) , f_a(0)]\) for (almost all) \(a \in E\). They also provide an argument showing that \(E\) is in the closure of the set of parameter values \(a\) for which some iterate of the critical point hits the repelling fixed point.