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.
View ims08-03 (PDF format)