Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

2008/04/14 by Artur Avila, Avila, Artur, Mikhail Lyubich +3
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.0804.2197

openalex publication_date 2008/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the parameter space of unicritical polynomials f_c:z↦ z^d+c. For complex parameters, we prove that for Lebesgue almost every c, the map f_c is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map f_c is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the ``principal nest'' of parapuzzle pieces.

Citations

• On Fibers and Renormalization of Julia Sets and Multibrot Sets

Related

• Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures
• Near parabolic renormalization for unicritical holomorphic maps
• Parabolic limits of renormalization
• Phase-parameter relation and sharp statistical properties for general families of unimodal maps
• Combinatorial rigidity for unicritical polynomials