Abstract
We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ 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.
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@article {ALM, MRKEY = {2018784},
AUTHOR = {Avila, Artur and Lyubich, Mikhail and de Melo, Welington},
TITLE = {Regular or stochastic dynamics in real analytic families of unimodal maps},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {154},
YEAR = {2003},
NUMBER = {3},
PAGES = {451--550},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {37E05 (30D05 37C15 37E20 37F45)},
MRNUMBER = {2006i:37083},
MRREVIEWER = {Feliks Przytycki},
ZBLNUMBER = {1050.37018},
} -
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@misc{ ALS,
author={Avila, A. and Lyubich, M. and Shen, W.},
TITLE={Parapuzzle of the Multibrot set and typical dynamics of unimodal maps},
JOURNAL={J. European Math. Soc.},
NOTE={to appear},
} -
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BOOKTITLE = {Topological methods in modern mathematics},
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ADDRESS = {Cambridge},
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@incollection {R, MRKEY = {1765086},
AUTHOR = {Roesch, Pascale},
TITLE = {Holomorphic motions and puzzles (following {M}. {S}hishikura)},
BOOKTITLE = {The {M}andelbrot set, theme and variations},
SERIES = {London Math. Soc. Lecture Note Ser.},
NUMBER = {274},
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PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2000},
MRCLASS = {37F45 (30D05 37F10 37F50)},
MRNUMBER = {2001c:37046},
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@incollection{Sch1, MRKEY = {1755449},
AUTHOR = {Schleicher, D.},
TITLE = {Rational parameter rays of the {M}andelbrot set},
BOOKTITLE = {G{é}om{é}trie Complexe et Syst{è}mes Dynamiques (Orsay, 1995)},
JOURNAL = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {261},
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PAGES = {405--443},
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@incollection{Sch2,
author = {Schleicher, Dierek},
TITLE={On fibers and renormalization of Julia sets and Multibrot sets},
BOOKTITLE={Geometry and Applications: A Jubilee of Benoit Mandelbrot},
VOLUME=1, pages={477--517},
SERIES={Proc. Sympos. Pure Math.},
number={72},
PUBLISHER={Amer. Math. Soc.},
ADDRESS={Providence, RI},
YEAR={2004},
NOTE={and Preprint IMS at Stony Brook \# 13 (1998)},
}
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