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 Pc : z t- x2 + c, 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 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.

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