Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

Avila, Artur; Lyubich, Mikhail; Shen, Weixiao

Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

Authors

Avila, Artur

Lyubich, Mikhail

Shen, Weixiao

Description

We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. 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.