Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

Details Parapuzzle of the Multibrot set and typical dynamics of unimodal maps Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

2008-04-14

arxiv.org/abs/0804.2197v1

Mathematics

Dynamical Systems

Scientific paper

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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