We study the parameter space of unicritical polynomials fc : z ↦ zd + 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.
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Artur Avila, Mikhail Lyubich, Weixiao Shen, Parapuzzle of the multibrot set and typical dynamics of unimodal maps. J. Eur. Math. Soc. 13 (2011), no. 1, pp. 27–56DOI 10.4171/JEMS/243