Almost every real quadratic map is either regular or stochastic

In this paper we complete a program to study measurable dynamics in the real quadratic family. Our goal was to prove that almost any real quadratic map P_c : z → x² + c, c ∈ [-2, 1/4], has either an attracting cycle or an absolutely continuous invariant measure. The final step, completed here, is to prove that the set of infinitely renormalizable parametric values c ∈ [-2, 1/4] has zero Lebesgue measure. We derive this from a Renormalization Theorem which asserts uniform hyperbolicity of the full renormalization operator. This theorem gives the most general real version of the Feigenbaum-Coullet-Tresser universality, simultanuously for all combinatorial types.