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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 → 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.
Mikhail Lyubich (Mon,) studied this question.