JournalsjemsVol. 13 , No. 1DOI 10.4171/jems/243

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

  • Artur Avila

    Université Pierre et Marie Curie, Paris, France
  • Mikhail Lyubich

    Stony Brook University, United States
  • Weixiao Shen

    University of Scinece and Technology of China, Hefei, China
Parapuzzle of the multibrot set and typical dynamics of unimodal maps cover

Abstract

We study the parameter space of unicritical polynomials fc : zzd + c. For complex parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map fc 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.