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Local connectivity of Julia sets for unicritical polynomials. (English) Zbl 1180.37072
Yoccoz proved that if a quadratic polynomial is at most finitely many renormalizable and if all periodic points are repelling, then the Julia set is locally connected. The main result of this paper is to extend this result to polynomials $$z\mapsto z^d+c$$ where $$d\geq 2$$. A major tool is a covering lemma proved by the same authors [Ann. Math. (2) 169, No. 2, 561–593 (2009; Zbl 1203.30011)].

MSC:
 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:
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