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Local connectivity of Julia sets for unicritical polynomials

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Jeremy Kahn, Mikhail Lyubich

Dynamical Systems · Submitted 10 May 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected.

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arXiv IDmath/0505194 (version 1)
Pith IDEJQK37YY, Pith's permanent record of this paper and its review
CitationJeremy Kahn, Mikhail Lyubich. "Local connectivity of Julia sets for unicritical polynomials." arXiv:math/0505194 (2005). https://pith.science/paper/EJQK37YY
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,024 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.91 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in math.DS/0505191 that give control of moduli of annuli under maps of high degree.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. result 0.94 0.96 0.96 asserted 1.00
s1 (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. background 0.61 0.35 0.10 asserted 0.99
s2 The proof of a priori bounds makes use of new analytic tools developed in math.DS/0505191 that give control of moduli of annuli under maps of high degree. method 0.96 0.57 0.02 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in math.DS/0505191 that give control of moduli of annuli under maps of high degree.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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  • There has been a growth of interest in the dynamics and parameter spaces of degree $d$ unicritical polynomials (Avila, Kahn and Lyubich [ref] , Milnor [ref] , Chéritat [ref] ) in the recent years.

    Rational Parameter Rays of The Multibrot Sets · Introduction · 0.97

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