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37-XX Dynamical systems and ergodic theory
37Fxx Complex dynamical systems
37F35 Conformal densities and Hausdorff dimension

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Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

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.

Porosity of Julia sets of non-recurrent and parabolic Collet-Eckmann rational functions.

Przytycki, F., Urbański, M. (2001)

Annales Academiae Scientiarum Fennicae. Mathematica

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