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arXiv · 0804.2197

Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

Artur Avila·Mikhail Lyubich·Weixiao Shen
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Abstract

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.

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Artur Avila, Mikhail Lyubich, Weixiao Shen. 2008-04-14. Parapuzzle of the Multibrot set and typical dynamics of unimodal maps. https://arxiv.org/abs/0804.2197

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