Geodesic
Local connectivity of Julia sets for unicritical polynomials
Jeremy Kahn1 ; Mikhail Lyubich2
1Department of Mathematics<br/>Stony Brook University<br/>Stony Brook, NY 11794-3651<br/>United States
2Department of Mathematics<br/>University of Toronto<br/>40 St. George Street<br/>Toronto, ON M5S 2E4<br/>Canada
Annals of mathematics, Tome 170 (2009) no. 1, pp. 413-426
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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.

DOI : 10.4007/annals.2009.170.413

Jeremy Kahn  1   ; Mikhail Lyubich  2

1 Department of Mathematics<br/>Stony Brook University<br/>Stony Brook, NY 11794-3651<br/>United States
2 Department of Mathematics<br/>University of Toronto<br/>40 St. George Street<br/>Toronto, ON  M5S 2E4<br/>Canada 
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Jeremy Kahn; Mikhail Lyubich. Local connectivity of Julia sets for unicritical polynomials. Annals of mathematics, Tome 170 (2009) no. 1, pp. 413-426. doi: 10.4007/annals.2009.170.413

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