Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila; Mikhail Lyubich; Weixiao Shen

doi:10.4171/JEMS/243

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila
, Mikhail Lyubich
, Weixiao Shen

Mathematical Science Institute

Institut de Mathématiques de Jussieu-Paris Rive Gauche
Instituto National de Matemática Pura e Aplicada
University of Science and Technology of China
National University of Singapore

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

We study the parameter space of unicritical polynomials f_c: z 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.

Original language	English
Pages (from-to)	27-56
Number of pages	30
Journal	Journal of the European Mathematical Society
Volume	13
Issue number	1
DOIs	https://doi.org/10.4171/JEMS/243
State	Published - 2011

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abstract = "We study the parameter space of unicritical polynomials fc: z zd + c. For complex parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map fc 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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AU - Lyubich, Mikhail

AU - Shen, Weixiao

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N2 - We study the parameter space of unicritical polynomials fc: z zd + c. For complex parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map fc 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.

AB - We study the parameter space of unicritical polynomials fc: z zd + c. For complex parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map fc 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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DO - 10.4171/JEMS/243

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JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

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ER -

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Abstract

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