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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)].

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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[1] B. Branner and J. H. Hubbard, ”The iteration of cubic polynomials, II: Patterns and parapatterns,” Acta Math., vol. 169, iss. 3-4, pp. 229-325, 1992. · Zbl 0812.30008 · doi:10.1007/BF02392761
[2] A. Douady and J. H. Hubbard, ”On the dynamics of polynomial-like mappings,” Ann. Sci. École Norm. Sup., vol. 18, iss. 2, pp. 287-343, 1985. · Zbl 0587.30028 · numdam:ASENS_1985_4_18_2_287_0 · eudml:82160
[3] J. H. Hubbard, ”Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz,” in Topological Methods in Modern Mathematics, Houston, TX: Publish or Perish, 1993, pp. 467-511. · Zbl 0797.58049
[4] J. Kahn and M. Lyubich, ”The quasi-additivity law in conformal geometry,” Ann. of Math., vol. 169, iss. 2, pp. 561-593, 2009. · Zbl 1203.30011 · doi:10.4007/annals.2009.169.561 · annals.math.princeton.edu · arxiv:math/0505191
[5] M. Lyubich, ”Dynamics of quadratic polynomials. I, II,” Acta Math., vol. 178, iss. 2, pp. 185-247, 247, 1997. · Zbl 0908.58053 · doi:10.1007/BF02392694
[6] G. Levin and S. van Strien, ”Local connectivity of the Julia set of real polynomials,” Ann. of Math., vol. 147, iss. 3, pp. 471-541, 1998. · Zbl 0941.37031 · doi:10.2307/120958 · arxiv:math/9504227
[7] J. Milnor, ”Local connectivity of Julia sets: expository lectures,” in The Mandelbrot Set, Theme and Variations, Lei, T., Ed., Cambridge: Cambridge Univ. Press, 2000, pp. 67-116. · Zbl 1107.37305 · arxiv:math/9207220
[8] J. Milnor, ”Periodic orbits, externals rays and the Mandelbrot set: an expository account,” in Géométrie complexe et systèmes dynamiques: Colloque en l’honneur d’Adrien Douady, Paris: Soc. Mat. de France, 2000, pp. 277-333. · Zbl 0941.30016
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