## Institute for Mathematical Sciences

## Preprint ims08-03

** Artur Avila, Mikhail Lyubich and Weixiao Shen**
*Parapuzzle of the Multibrot set\\ and typical dynamics of unimodal maps*

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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