2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154105We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite) renormalizations.57 pages. In this update, we added lots of figures and detailed the proofs, especially in the section called "the control of the post critical set"Dynamical Systems37FXXtext