# Every real quadratic map is either regular or stochastic

### Abstract

In this paper we complete a program to study measurable dynamics in the real quadratic faily. Our goal was to prove that almost any real quadratic map $P_c \colon z\mapsto x^2 +c$, $c\in [-2,1/4]$, 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 $c\in [-2,1/4]$ 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 univerality simultanuously for all combinatorial types.

Mikhail Lyubich