Almost Every Real Quadratic Map Is Either Regular or Stochastic

Mikhail Lyubich
2002 Annals of Mathematics  
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 → x 2 + 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
more » ... erbolicity of the full renormalization operator. This theorem gives the most general real version of the Feigenbaum-Coullet-Tresser universality, simultanuously for all combinatorial types. 1 {λ} stand for the vertical fibers of a bidisk V, where π 1 : V → Λ is the natural projection. We will assume that they are quasi-disks containing 0. Denote by ∂ h V = ∪ λ∈Λ ∂V λ the horizontal boundary of V. By definition, a map f : V → V between two bidisks V ⊂ V over Λ is called a quadratic-like family over Λ = Λ f if f is a holomorphic endomorphism preserving the fibers such that every fiber restriction f λ : . ., is a normalized quadratic-like map with the critical point at 0. Clearly any quadratic-like family f represents a holomorphic curve in Q. We will use the same notation, f , for this curve. Let us say that a quadratic-like family f : V → V over (Λ, * ) is equipped if the base map f * is equipped with a tubing H * (see (2.2)) and there is an
doi:10.2307/3597183 fatcat:l67zlaqtg5fxfdh7sqnlya676i