# Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila; Mikhail Lyubich; Weixiao Shen

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 1, page 27-56
- ISSN: 1435-9855

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topAvila, 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 -

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