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 Principal Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in [KL09] that give control of moduli of annuli under maps of high degree.
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[BH]
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AUTHOR = {Branner, Bodil and Hubbard, John H.},
TITLE = {The iteration of cubic polynomials, {II}: {P}atterns and parapatterns},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {169},
YEAR = {1992},
NUMBER = {3-4},
PAGES = {229--325},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {30D05},
MRNUMBER = {94d:30044},
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DOI = {10.1007/BF02392761},
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JOURNAL = {Ann. Sci. École Norm. Sup.\/},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {18},
YEAR = {1985},
NUMBER = {2},
PAGES = {287--343},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {58F11 (30D05)},
MRNUMBER = {87f:58083},
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ZBLNUMBER = {0587.30028},
URL = {http://www.numdam.org/item?id=ASENS_1985_4_18_2_287_0},
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@incollection {H, MRKEY = {MR1215974},
AUTHOR = {Hubbard, J. H.},
TITLE = {Local connectivity of {J}ulia sets and bifurcation loci: three theorems of {J}.-{C}. {Y}occoz},
BOOKTITLE = {Topological Methods in Modern Mathematics},
venue={Stony {B}rook, 1991},
pages={467--511},
PUBLISHER = {Publish or Perish},
ADDRESS = {Houston, TX},
YEAR = {1993},
MRCLASS = {58F23 (28A80 30C10 58F12)},
MRNUMBER = {94c:58172},
MRREVIEWER = {Christoph Bandt},
ZBLNUMBER = {0797.58049},
} -
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@article {coveringlemma, MRKEY = {2480612},
AUTHOR = {Kahn, Jeremy and Lyubich, Mikahil},
TITLE = {The quasi-additivity law in conformal geometry},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {169},
YEAR = {2009},
NUMBER = {2},
PAGES = {561--593},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {37Fxx},
MRNUMBER = {2480612},
DOI = {10.4007/annals.2009.169.561},
} -
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@article {puzzle, MRKEY = {MR1459261},
AUTHOR = {Lyubich, Mikhail},
TITLE = {Dynamics of quadratic polynomials. {I},
{II}},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {178},
YEAR = {1997},
NUMBER = {2},
PAGES = {185--247, 247--297},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {58F23 (30C10 30D05)},
MRNUMBER = {98e:58145},
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ZBLNUMBER = {0908.58053},
DOI = {10.1007/BF02392694},
} -
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@article {LS, MRKEY = {MR1637647},
AUTHOR = {Levin, Genadi and van Strien, Sebastian},
TITLE = {Local connectivity of the {J}ulia set of real polynomials},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {147},
YEAR = {1998},
NUMBER = {3},
PAGES = {471--541},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {58F23 (30D05 58F03)},
MRNUMBER = {99e:58143},
MRREVIEWER = {Jacek Graczyk},
ZBLNUMBER = {0941.37031},
DOI = {10.2307/120958},
} -
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@incollection {M, MRKEY = {MR1765085},
AUTHOR = {Milnor, John},
TITLE = {Local connectivity of {J}ulia sets: expository lectures},
BOOKTITLE = {The {M}andelbrot Set, Theme and Variations},
SERIES = {London Math. Soc. Lecture Note Ser.},
number = {274},
PAGES = {67--116},
editor = {Tan Lei},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2000},
MRCLASS = {37F50 (30D05 37-02)},
MRNUMBER = {2001b:37073},
ZBLNUMBER = {1107.37305},
} -
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@incollection{M-rays, MRKEY = {MR1755445},
AUTHOR = {Milnor, John},
TITLE = {Periodic orbits, externals rays and the {M}andelbrot set: an expository account},
booktitle = {G{é}om{é}trie complexe et syst{è}mes dynamiques: Colloque en l'honneur d'Adrien Douady},
venue={Orsay, 1995},
series = {Astérisque},
FJOURNAL = {Astérisque},
NUMBER = {261},
publisher = {Soc. Mat. de France},
address={Paris},
YEAR = {2000},
PAGES = {277--333},
ISSN = {0303-1179},
MRCLASS = {37F45 (30D05)},
MRNUMBER = {2002e:37067},
MRREVIEWER = {Peter Ha{ï}ssinsky},
}
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