Almost every real quadratic map is either regular or stochastic

1997/07/15 by Mikhail Lyubich, Lyubich, Mikhail · 2 citations
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.math/9707224

openalex publication_date 1997/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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↦ x²+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.

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• Unfolding chaotic quadratic maps --- parameter dependence of natural measures
• Rigorous computation of expansion in one-dimensional dynamics

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