Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Avila, Artur, Lyubich, Mikhail, and Shen, Weixiao. "Parapuzzle of the multibrot set and typical dynamics of unimodal maps." Journal of the European Mathematical Society 013.1 (2011): 27-56. <http://eudml.org/doc/277598>.

@article{Avila2011,
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.},
author = {Avila, Artur, Lyubich, Mikhail, Shen, Weixiao},
journal = {Journal of the European Mathematical Society},
keywords = {unimodal maps; parameter spaces; hyperbolic maps; renormalizable maps; Collet-Eckmann maps; unimodal maps; parameter spaces; hyperbolic maps; renormalizable maps; Collet-Eckmann maps},
language = {eng},
number = {1},
pages = {27-56},
publisher = {European Mathematical Society Publishing House},
title = {Parapuzzle of the multibrot set and typical dynamics of unimodal maps},
url = {http://eudml.org/doc/277598},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Avila, Artur
AU - Lyubich, Mikhail
AU - Shen, Weixiao
TI - Parapuzzle of the multibrot set and typical dynamics of unimodal maps
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 1
SP - 27
EP - 56
AB - 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.
LA - eng
KW - unimodal maps; parameter spaces; hyperbolic maps; renormalizable maps; Collet-Eckmann maps; unimodal maps; parameter spaces; hyperbolic maps; renormalizable maps; Collet-Eckmann maps
UR - http://eudml.org/doc/277598
ER -