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Almost every real quadratic map is either regular or stochastic. (English) Zbl 1160.37356
Summary: We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family \(P_c: x\mapsto x^2+c\) has zero measure. This yields the statement in the title (where “regular” means to have an attracting cycle and “stochastic” means to have an absolutely continuous invariant measure). An application to the MLC problem is given.

MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
28D05 Measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
37E20 Universality and renormalization of dynamical systems
37F25 Renormalization of holomorphic dynamical systems
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