Combinatorial rigidity for unicritical polynomials

Artur Avila; Jeremy Kahn; Mikhail Lyubich; Weixiao Shen

doi:10.4007/annals.2009.170.783

Combinatorial rigidity for unicritical polynomials

Artur Avila
, Jeremy Kahn
, Mikhail Lyubich
, Weixiao Shen

Mathematical Science Institute

Stony Brook University

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

31 Scopus citations

Abstract

We prove that any unicritical polynomial f_c : z → z^d + C which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the "Multibrot set") is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.

Original language	English
Pages (from-to)	783-797
Number of pages	15
Journal	Annals of Mathematics
Volume	170
Issue number	2
DOIs	https://doi.org/10.4007/annals.2009.170.783
State	Published - 2009

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10.4007/annals.2009.170.783

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title = "Combinatorial rigidity for unicritical polynomials",

abstract = "We prove that any unicritical polynomial fc : z → zd + C which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the {"}Multibrot set{"}) is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.",

author = "Artur Avila and Jeremy Kahn and Mikhail Lyubich and Weixiao Shen",

year = "2009",

doi = "10.4007/annals.2009.170.783",

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publisher = "Department of Mathematics at Princeton University",

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T1 - Combinatorial rigidity for unicritical polynomials

AU - Avila, Artur

AU - Kahn, Jeremy

AU - Lyubich, Mikhail

AU - Shen, Weixiao

PY - 2009

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N2 - We prove that any unicritical polynomial fc : z → zd + C which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the "Multibrot set") is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.

AB - We prove that any unicritical polynomial fc : z → zd + C which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the "Multibrot set") is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.

UR - https://www.scopus.com/pages/publications/71449110340

U2 - 10.4007/annals.2009.170.783

DO - 10.4007/annals.2009.170.783

M3 - Article

SN - 0003-486X

VL - 170

SP - 783

EP - 797

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -

Combinatorial rigidity for unicritical polynomials

Abstract

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